Symptoms And Symptoms Of Acute Kidney Infection - 955 Words

During the second set of final clinical practice, I provided nursing care to the client with query sepsis and clostridium difficile as admitting diagnosis. The client had a history (Hx) of acute kidney infection (AKI) which led to dialysis. After resolving AKI, the patient went home, but soon returned to the hospital with severe diarrhea (5-6 episodes per day), confusion and symptoms of sepsis. Upon initial assessment, I found the patient oriented to name only, confused and lethargic, incontinent of urine and stool. The patient had bilateral crackles throughout the lung fields, gurgles upon exertion and tachypnea with respiratory rate 24-28. SpO2 level was within normal limits. The patient’s family reported that the current patient’s cognitive condition function was different from the baseline. The patient was difficult to arouse, with Glasgow Coma Scale (GCS) score 12-13. The heart rate was within normal limits, strong, irregular. Bilateral edema 2+ was present in lower legs, skin was warm to touch, pedal pulses palpable. The patient was on caloric count due to poor caloric intake. The family was frustrated because of recurrent hospitalization due to hospital-acquired infection and very concerned about possibility of poor outcome for the patient due to rapidly deteriorating general condition. My primary concern was the possible aspiration as evidenced by gurgles upon exertion. I elevated the head of bed and made sure that suctioning equipment was in place andShow MoreRelatedAcute Syndrome : Acute Nephritic Syndrome1165 Words   |  5 PagesAcute Nephritic Syndrome Introduction Acute nephritic syndrome is a group of symptoms that occurs with a few disorders that cause glomerulonephritis or swelling and inflammation of the glomeruli in the kidney as shown in Figure 1. Inflammation of the kidneys and glomerulus affects the function of the glomerulus – part of the kidney that filters blood, resulting in blood and protein to appear in urine – excess fluid also builds up in the body. Swelling of the body occurs when blood loses albuminRead MoreAcute Renal Failure Of The Urinary System1497 Words   |  6 Pages Diana Galeana MED 2049 Acute Renal Failure Instructor Michelle Earxsion- Lamothe 7/25/2014 Acute Renal Failure Although the function of the urinary system is used to filter and eliminate waste from the body, it also contributes with maintenance of homeostasis of water and blood pressure, regulates electrolytes, pH balance, and activates vitamin D. The urinary system consists of 2 kidneys which extract wastes from the blood, balance body fluid, and converts it into urineRead MoreEssay on Renal Failure1218 Words   |  5 Pagesand symptoms, the dietary modifications a nurse should teach, the medical management of acute renal failure, and finally the short and long term goals a nurse should make for their client. There are three causes of acute renal failure; prerenal causes, renal causes, and post renal causes. Prerenal causes are due to such factors as dehydration i.e... vomiting diarrhea, or sweating, or poor fluid intake. Other factors could also include weak or irregular blood flow to and from the kidneys becauseRead MoreAcute Renal Failure Essay example1093 Words   |  5 Pages Acute renal failure, also known as acute kidney injury is described to be a rapid loss of kidney function, or a rapid decline in renal filtration function. 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The Kidneys The role of the kidneys is to remove nitrogenous wastes from the body. Nitrogenous means that it is rich in the element nitrogen. Nitrogen in high concentrations in the body can cause several problems such as joint pain, strokes or heart attacks. The kidney is made up three parts: the renal cortex, the renal medulla and the renal pelvis. All mammals have two kidneys. The kidneys primary function is to regulate various body fluidsRead MoreThe Issues Associates with Acute Renal Failure1025 Words   |  4 PagesAcute renal failure is the most common kidney disease that exists today. It occurs when blood flow to the kidneys is in some way compromised which causes a sudden stop in kidney function. Acute renal failure is a very serious complication for a already hospitalized patient since they are already in a vulnerable state from staying in the hospital, in fact, it is the most common cause of death amongst hospitalized patients, and most commonly they occur because of a hospital worker s error. Acute renalRead MoreCauses And Treatment Of A Kidney1121 Words   |  5 PagesPyelonephritis As kidney is one of the very important organ of the body, its dysfunction may result in the fatal consequences. Unlike any other disruption to the kidney, Pyelonephritis may introduce some serious problem to the elderly. It is an inflammation of the kidney. Though this disease is not prone to old people, most chronic cases occur in people over 60 years of age. 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Essay Operant Conditioning - 1743 Words

Ivan Pavlov Classical conditioning is a method used in behavioral studies. It is known as classical because it is the first study of laws of learning/conditioning, It is a learned reaction that you do when evoked by a stimulus. Ivan Pavlov was the scientist who discovered classical conditioning. Ivan Pavlov was born in Russia. He lived from 1849 - 1936 . Pavlov’s field of study was physiology and natural science. One of Pavlov’s discoveries was the conditioning of dogs. While working with the dogs he investigated the salivation reaction when food was present. He discovered that the dogs would salivate when he entered the room with and without food. Pavlov then went on to training the dogs. He would ring a bell†¦show more content†¦The animal would response. It would then be rewarded. The response is then learned. If the response was not rewarded then it would disappear over time. The animals would be placed into the puzzle boxes and would only be able to escape by making certain responses such as pushing a button. The experiment with the cats entailed placing a hungry cat into the puzzle box. He then observed its behaviour. The cat would try and escape in order to obtain food. The cat in most cases would act on a â€Å"trial and error† method. When the cat obtained the food the trial and error behaviour decreased and the cat soon learnt how to escape quickly. The lear nt behaviour took time and did not improve quickly. The amount of time the cat spent in the box slowly shorted. Upon completion of his experiment Thorndike learnt that certain stimuli and responses became associated or dissociated from each other. He developed his law of effect. Thorndike concluded that animals learn by trial and error, or most importantly by reward and punishment. He then linked the behavior of the cats to all beings. His work with animals was the founding principle of Instrumental Learning. His most famous work was on the learning theory that lead to the development of operant conditioning by a psychologist called Burrhus.Frederic Skinner. Skinner was born in Pennsylvania on March 20, 1904. Skinners first studied English at Hamilton College. He often wrote for the college newspaper. HeShow MoreRelatedClassical or Operant Conditioning Essay1115 Words   |  5 PagesClassical conditioning developed from the findings of Ivan Pavlov, laying the foundations for behaviourism. From this J.B Watson and other behaviourists argued psychology should be indicative of predicting and controlling overt behaviour using the conditional reflex. (Watson, 1994). This essay will describe the important features of classical conditioning, consider their use in explaining pathological behaviour and will be answered using empirical evidence. The earlier part of the essay will focusRead MoreClassical and Operant Conditioning Essay1000 Words   |  4 Pagesworld. Classical conditioning and operant conditioning are both basic forms of learning, they have the word conditioning in common. Conditioning is the acquisition of specific patterns of b ehavior in the presence of well-defined stimuli. Classical conditioning is a type of learning in which an organism learns to transfer a natural response from one stimulus to another, previously neutral stimulus. Classical conditioning is achieved by manipulating reflexes. Operant conditioning is a type of learningRead MorePsy 390 Operant Conditioning Essay851 Words   |  4 PagesOperant Conditioning Dena Couch PSY 390 July 30, 2012 Dr. Thauberger Operant Conditioning In this paper there will be an examination of the Operant Conditioning theory. It will describe the theory, and compare and contrast the positive and negative reinforcement. It will determine which form of reinforcement is the most effective, and will give an explanation of the reasoning behind that choice. It will also give a scenario in which operant conditioning is applied and how it shapes behaviorRead MoreSkinners Operant Conditioning Theory Essay658 Words   |  3 PagesSkinners Operant Conditioning Theory B.F Skinner (1904-1990), an American psychologist who was the leading exponent of the school of psychology know as behaviourism, maintained the idea that learning is a result of any change in overt behaviour. Changes in behaviour are determined by the way an individual responds to events (stimuli) in the environment. Skinner described this phenomenon as operant conditioning. Action on part of the learner is called a response. WhenRead MoreOperant Conditioning in the Criminal Ju Essay2660 Words   |  11 Pagesï » ¿ Operant Conditioning in the Criminal Justice System Z. M. Keys Psychology of Criminal Behavior CCJS 461 17 October 2014 The only way to tell whether a given event is reinforcing to a given organism under given conditions is to make a direct test. We observe the frequency of a selected response, then make an event contingent upon it and observe any change in frequency. If there is a change, we classify the event as reinforcing to the organism under the existing conditionsRead MoreEssay on Comparison of Classical and Operant and Conditioning660 Words   |  3 Pagesbeing either classical conditioning or operant conditioning when we are dealing with Psychology terms. These two habituation methods are very comparable in nature, but do possess very specific distinctions in their differences. The major difference between classical and operant conditioning is the type of behaviors being conditioned. Classical is focused more on reflex and automatic actions whereas operant deals more with voluntary actions. Classical and operant conditioning are also different inRead MoreApplying Operant Conditioning to Hum an Behaviour Essay591 Words   |  3 PagesApplying Operant Conditioning to Human Behaviour Operant conditioning is when a way of learning by consequence. To put it basic, an action which is rewarded is more likely to be repeated, along with an action that is punished is less likely to be repeated. To apply this to an example of human behaviour, young children may have shaped behaviour due to operant conditioning; where desireable behaviour is rewarded (e.g. by giving a toy) the behaviour is being positivelyRead MoreB. F. Skinners Philosophy of Operant Conditioning Theory Essay591 Words   |  3 PagesI think that B.F. Skinner shares my philosophy in the behavioral aspects of education. There are many points that have expanded my philosophy. One was the operant conditioning theory, which is when the behavior is changed through positive and negative consequences depending on one’s behavior. 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An Email to a Friend Essay Example For Students

An Email to a Friend Essay You won’t believe how different this job turned out to be. I think credibility is the biggest thing I’ve struggled with so far. I always want to make sure that everything I say is correct and honest. After all, I’m trying to build my brand and if I ever want to be the next Anderson Cooper, I should at the very least be credible. In searching for the truth I often have to check, double check, and then cross-check references to ensure accuracy and coherency. It is incredible how many websites are less than helpful with this, as too many reporters can come off biased or try to put their own spin on a story. I think the key is to ensure that I am doing this right is to make sure I don’t take sides as well as stay truthful. Another issue I’m having is the influence that media has over people. Not everyone is media literate, so integrity becomes an even heavier burden because who knows how many people I’m influencing. How many people are looking at me as a source of truth or knowledge and how are they interpreting what I’m saying? Are they hearing my words as a call to arms or as a warning? Should I publish this story at all? I never thought that it would be so difficult to just report the facts. Hah! I recently wrote a story for a smaller newspaper that was clearly running low on ideas because it was something silly that happened in a county courthouse. A young woman with an ethnically specific last name made an honest mistake and brought a knife along in a diaper bag to a courthouse. She didn’t realize it was illegal to have it on her, and I think she really meant no harm. I reported just the facts: her name, her age, and what she was arrested for. There was nothing to interpret. She forgot it was there and handed it over without thinking twice about it according to the police, who didn’t even want to make this that big of a deal. Her baby couldn’t be more than six months old, she was so tiny, and according to the mother, it was given as a gift from the father. It had been his prized possession. Men. Haha! However, the comments for this article all but cried for this young lady’s blood. People were commenting on how she was an illegal immigrant who just wanted to get one over on the system and probably spoke no English. How absolutely absurd is that? What did they have to go off of? A last name that wasn’t â€Å"Smith†? Ridiculous! And yet, this is what I have to be careful of: People who already made up their minds without doing an iota of research. It’s tough and sad, and sometimes, I don’t know how to combat it. Maybe experience will teach me that this story could have gone unknown. Anyway, that’s all I have to report for now. Hopefully, we can meet up soon! Your friend, Arnold

Pre AND Post 1900 Comparative Essay Example Essay Example

Pre AND Post 1900 Comparative Essay Example Paper Pre AND Post 1900 Comparative Essay Introduction In this essay, I have chosen to compare and contrast the following two poems – ‘Even Tho’ by Grace Nichols and ‘To his Coy Mistress’ by Andrew Marvell. These two poems were roughly written three centuries apart, and so the social factors and religious beliefs as well as other things were different. Therefore different ideas would be portrayed in the poems.In the 17th century when ‘To his Cot Mistress’ was written, women were not known for writing poems about love let alone sex, as women’s opinions weren’t respected. If a woman was sent a poem like the one that Andrew Marvell wrote, then she would probably go weak at the knees and do whatever was asked of her. However, in the late 20th century, when ‘Even Tho’ was written, women’s opinions and rights were respected in society and more and more female poets emerged with some ideas that were once thought unacceptable for women.One other main factor that affe cted the ideas portrayed in these poems is partly to do with religion and partly to do with what used to be thought of as socially acceptable and what is thought as socially acceptable now. At the time ‘To his Coy Mistress’ was written, women were expected to keep their virginity until they were married. This is why the poem is set out as an argument, trying to persuade his lover to have sex with him. This is different to the time when ‘Even Tho’ was written because women were not expected to stay a virgin.I will now discuss the similarities and differences between the content of the first poem, ‘To his Coy Mistress’ and the second poem ‘Even Tho.’ The first poem ‘To his Coy Mistress’ is unusual for the time as it has an untraditional structure. It has no verses but it has three sections each with a different number of lines. The fact that it has no verses suggests that the subject of the poem never really changes, onl y the perspective of the poet. This is so, that the poet can present an effective argument. In the poem, some of the sentences carry on to the next line – this is to increase the pace and build up a good argument. Another reason for why it is unusual for its time is because of the purpose. At the time that the poem was written, women were expected to keep their virginity until they were married, but the poet is asking his lover to lose her virginity, although they aren’t married.The three different sections separate the different parts of the argument that he is putting across. In the first section, we see the poet describing what he would do if he could spend eternity with his lover, A hundred years should go to praise thine eyes, and on thy forehead gaze.’ In the second section, he is saying that although it would be nice to do the things which he said in paragraph one, he can’t because he wont live forever and she will eventually die with her ‘qu aint honour.’ A quote to show this would be, ‘Nor, in thy marble vault, shall sound my echoing song: then worms shall try that long preserved virginity.’ The final section concludes the argument by saying what they should do in order that his mistress does not die having never ‘expressed her love’ towards her lover.I think, that from the perspective of the poet’s mistress, the poem is quite successful. This is because the poet uses some very effective methods to persuade his lover. In the first section, he sweet-talks his lover before scaring her into bed with thoughts of death in the second section. This is why the structure is very good as in my opinion, it allows the poet to separate the poem into three separate parts, creating an effective argument.The second poem, ‘Even Tho’ by Grace Nichols is in many ways very different. Her ideas would have been seen as very controversial if they were expressed at the same time as ‘T o his Coy Mistress’ was written. We can see this from the purpose of the poem, the intention being to inform her lover that she wants sex without the commitment. This is unlike the woman in ‘To his Coty Mistress’ as she has to be heavily persuaded to do such a thing. At the time ‘Even Tho’ was written, women were far more in control of their own lives and didn’t succumb to obsequiousness as easily.A way that these two poems are similar is that they both have an untraditional structure. ‘Even Tho,’ is very much a free verse poem, in that it doesn’t have a set number of lines per verse or a set number of words per line. It also has very little punctuation. This enables it to highlight the untraditional ideas it contains. This is similar to the reasoning behind an unconventional structure in ‘To his Coy Mistress.’ Both of these poems are written in the first person, this means that both of the poets can reinforce their emotions by making the poem more personal. Again, ‘Even Tho’ and ‘To his Coy Mistress’ are partly similar in purpose; they both express emotion and give the poets view on their relationship.The manner in which the poets express their emotions are quite different. In ‘Even Tho,’ the poet uses a positive and light-hearted tone to put across her feelings, whereas in ‘To his Coy Mistress,’ Grace Nichols uses more traditional ideas of love as well as his unorthodox views on his relationship with his lover.In my opinion, both poems may well be seen as relatively offensive because they are very biased as they only take into account their own opinions. What about the other person in the relationship? They may desire something different! ‘To his Coy Mistress’ is especially offensive because the poet’s requirements are just sexual pleasure.As is to be expected, the language of these two poems is very different in ma ny ways. The vocabulary in ‘To his Coy Mistress’ is very dated as the poem was written in the mid 17th century. Unfamiliar words like ‘thou’ and ‘thine’ are used, which could be difficult to comprehend for the majority of people in this day and age.There is a lot of imagery used in this poem to create a picture in our minds of the poet’s lover and the relationship that they share. The poet uses metaphors such as ‘Times winged chariot’ and ‘Iron gates of life.’ He also uses similes, for example, ‘the youthful hue, sits on thy skin like morning dew.’ In addition to this simile, this sentence shows us another technique, which is called personification. All of these techniques, plus some very elaborate descriptions build up some very apparent images.The poem rhymes all the way through in couplets and this is one of the only regularities in this poem. This means that when you read the poem, as well as pi cking up the untraditional ideas, you also sense the regularity, which makes the poem easier to identify with. It also has the same alliteration in it, for example: ‘long love’ and ‘love at lower rate.’ These are both to be found in the first section, along with some assonance that gives similar sounds, for example: ‘should’st rubies find.’ These are all soft, sumptuous and loving sounds which comply with the messages in the first section. As you would think, the second section contains more hard sounds, for example: ‘turn to dust’ and the third section contains dramatic and indicative ideas and so uses sounds to back these up, for example: ‘instant fires’ and ‘rough strife.’The poem appeals to a couple of the senses, mainly sight because of all the imagery used and sound because of the descriptions used associated to sound. The beat of the poem is also regular, as it has approximately 8 – 10 syllables per line; this suggests that it flows when read.The vocabulary of ‘Even Tho’ is very different to that of ‘To his Coy Mistress,’ mainly because it was written much later, round about the 1970’s or 1980’s. This meant that the language was much more up to date. The time that it was written also means that it was possible for the poem to be written with a Caribbean dialect. An example of this is, ‘Keep to de motion,’ and ‘leh we go.’It was very unusual to see a poem written in a Caribbean dialect from the period in which ‘To his Coy Mistress’ was composed, so this highlights a difference between the two poems. The dialect allows the poet to bring in some of her ethnicity and culture to her work.A connection between the two poems is that they both use ample amounts of imagery. In ‘Even Tho’ metaphors are used, for example: ‘I’m all watermelon and star apple and plum when y ou touch me.’ This metaphor shows us what the poet is feeling. Grace Nichols uses juicy, soft fruits to describe it because that’s how she feels. The poem is very short, and due to this, we don’t find any similes or personification, but the poem does have some very interesting descriptions to create images, such as, ‘you be banana, I be avocado,’ which describes the male and female sex organs – the banana symbolizes the penis as it is very hard and long, whereas the avocado denotes a vagina as it is very warm, soft and in particularly red! This type of imagery is somewhat different to that used in ‘To his Coy Mistress’ as it is more intimate, light-hearted and humorous, unlike the romantic and sometimes frightening imagery of ‘To his Coy Mistress.’ The sounds to the poem are one of the keys to its success. Assonance such as ‘watermelon, strar apple and plum’ gives juicy and sumptuous sounds that appeal v ery much to the reader’s sense of taste and touch. The imagery used when talking about the male and female sex organs, ‘banana and avocado’ is quite amusing, and so appeals to the reader’s sense of sight.Besides the poem being outwardly funny, it has a relatively fundamental underlying message about the poet’s relationship, which is shown in the poet’s choice of repetition. The reiteration of ‘Even Tho’ and ‘leh we break free,’ is what tells the poet’s lover exactly what she wants from their relationship. She wants to be an individual ‘even tho’ she enjoys having fun and spending time with her lover.In my opinion, the poem is similar to ‘To his Coy Mistress’ when it comes to pace and rhythm as they are both irregular and so stressing their equally unorthodox messages.In conclusion, I would say that these poems aren’t completely unrelated, and the main thing that influences thei r differences is the time in which they were written. They both have similar purposes, only the perspective changes. It is largely male in ‘To his Coy Mistress,’ but incredibly female in ‘Even Tho. Another way that time has made the poems more different is the way that they are presented. ‘To his Coy Mistress’ is presented as an argument that is trying to persuade the poet’s lover to give in to her passion for the poet and lose her virginity.However, ‘Even Tho’ is more of a story than an argument. This is because she doesn’t feel that she needs to persuade men to do what she wants them to do, only tell them how she feels. Andrew Marvell felt that he needed to persuade his lover, as simply making a suggestion would not be enough. These are just a few examples of the ways that they are different, and of course, there are many more, but we must remember the simple similarities. Both poems are about sexual relationships, they ar e both written in the first person and to conclude, they both express emotions!My particular favourite out of the two poems has to be ‘Even Tho.’ The reason being, it is so simple, but yet has so many layers of meaning and tone. It is also comical as the imagery is fairly explicit as it talks about ‘bananas’ and ‘avocados’ representing the sexual organs on a human’s body. The poem ‘Even Tho’ also has a far more informal style of writing than ‘To his Coy Mistress’ and communicates to more than just the poet’s lover. All of this is why ‘Even Tho’ is my favourite poem out of the two. Pre AND Post 1900 Comparative Essay Thank you for reading this Sample!

Beowulf vs. The Patriot essays

Beowulf vs. The Patriot essays When looking at the comparison of the stories Beowulf and the Patriot, there are some definite differences, but there are also some things very common to both of these heroes. Lets have a look at some of the major differences and common points between the two characters. Beowulf was originally from Geatland, and once he heard the news of Grendel attacking Hrothgars kingdom, he packed his bags and headed for Danish lands. In other words, he fought for a country beside his own. The Patriot is much different, actually the exact opposite. The Patriot is defending his home country, fighting against people coming from the other country (Britian). If the Patriot came to American shores to fight, it would have been to fight against the Americans instead of for them. Another noticeable difference in the styles of the two heroes is how much help each gets when fighting. Beowulf fights every single battle by himself, and never seems to request for help. It seems as if he would rather lose a battle alone that win with the help of a friend. On the other hand, the Patriot is always searching for teammates to aid with the battles. Which brings up another difference, dealing with numbers. Beowulf is consistently matched up against no more than 1 enemy at a time. The Patriot is always outnumbered, no matter what the circumstances are, because his battle is a war. In Beowulfs case, it seems that everyone in the community he is trying to help is very friendly and supportive of his actions to defend the people of the community. On the other hand is the Patriot, who continually tries to prevent his community from entering into the war, mainly because if they did take part in war, his son would also join. Since almost everyone in the community is for going to war, and the Patriot is not, no one in the community really agrees or backs his decision. Moving on to similarities between the two ch ...

How to Use the Perl Array join() Function

How to Use the Perl Array join() Function The Perl programming language  join() function is used to connect all the elements of a specific list or array into a single string using a specified joining expression. The list is concatenated into one string with the specified joining element contained between each item. The syntax for the join() function is: join EXPR, LIST. Join() Function at Work In the following example code, EXPR uses three different values.  In one, it is a hyphen. In one, it is nothing, and in one, it is a comma and a space. #!/usr/bin/perl$string join( -, red, green, blue );printJoined String is $string\n;$string join( , red,  green,  blue  ); printJoined String is $string\n;$string  Ã‚  join(  , ,  red,  green,  blue  );printJoined String is $string\n; When the code is executed, it returns the following: Joined String is red-green-blueJoined String is redgreenblueJoined String is red, green, blue The EXPR is only placed between pairs of elements in LIST. It is not placed before the first element or after the last element in the string.   About Perl Perl,  which is an interpreted programming language, not a compiled language, was a mature programming language long before the web, but it became popular with website developers because most of the content on the web happens with text, and Perl is designed for text processing. Also, Perl is friendly and offers more than one way to do most things with the language.

World war one and US Essay Example | Topics and Well Written Essays - 750 words

World war one and US - Essay Example The main purpose of the war was struggling for division of the world. However, the United States wasn’t in hurry to enter the war. First of all, a conflict between the US and other countries hadn’t reached the extreme point that could lead to military actions; secondly, the majority of Americans didn’t think that interfering into European affairs (and vice versa) was a good idea. The third reason was that US army wasn’t ready for large-scale war, because it was formed on a voluntary basis and had approximately 100  000 badly trained soldiers. But in the 7th of May 1915 German submarine sank the British largest passenger liner â€Å"Luisitania†, including 128 Americans that were on board. A report about this event caused a storm of dissatisfaction in the United States. Despite multiple warnings, German repeatedly attacked American ships. President Woodrow Wilson was an advocate of nonintervention of USA into European war, but the actions of the Ger mans forced him to announce the transition to the US policy of armed neutrality (26th of February 1917), which meant a gap of diplomatic relations between the US and Germany. The further German aggression forced Wilson to enter the war (6 of April 1917) on Allies’ side, against Germany (Kennedy 46). For the first time in US history a law on conscription was passed. Losses suffered by Americans were relatively not that big (approximately 104,000 lives). For comparison, 26 million people were killed during the entire war, half of them were civilians. The war abruptly changed the balance of forces in world political arena. It provoked huge amount of revolutions: Bolsheviks came to power in Russia and formed their own government, Germany and Austria-Hungary collapsed and lost their influence. The military destruction and revolutions weakened Europe, while US entry into the war caused new rise of

Communal Supportive Action Theory Assignment Example | Topics and Well Written Essays - 500 words

Communal Supportive Action Theory - Assignment Example The initial part of the theory describes how people with measurable approach calculate their actions to benefit them self only, but they suffer long-term loneliness, lack of support and pleasure and disheveled reputation eventually. On the other hand, considerate individuals take a course of action which is less beneficial for them, but more beneficial towards larger humanity then they would earn long-term support, respect, a fiscal and sentimental advantage over the former one. The ‘Communal Supportive Action’ theory entails the idea of pleasantly surviving and sustaining in a society. This theory lays stress on the fact that individuals neither live alone nor their actions impact less. Hence, if they take actions according to their ease, preference, and lifestyle they would be withholding the notion of communal advantage. For instance, if garbage is thrown out of a house and dumped in a street, the trash would make the passage narrow, would produce long-term detrimental effects (health), would make a good home for pests and rats, would make the area look dirty and that would automatically evaluate the community. These self-centered actions are taken on the bases of personal priorities (shortage of time, lack of strength to walk the extra mile, lack of sense of cleanliness and lack of respect for others living in the surrounding). If one assesses the abovementioned theory with regard to objective approach then one realizes that Welfares, NGO’s, non-profitable organizations, religious and community beneficial services always concentrate on larger good instead of personal ones. Moreover, if individuals focus on their preferences then they attain short-term benefit and lack of dignity in the society as well.

The Handmaids Tale Essay Example for Free

The Handmaids Tale Essay This book is a depiction of an anti-utopian future society, along with others like 1984 and Brave New World. It combines a futuristic reality, feminism and politics to create a very detailed novel considering many different aspects of Gilead. Offred is the complex lead character who draws us into the seemingly perfect but corrupt world of Gilead. Her pain is experienced by the readers who long to remember exactly what she has forgotten, and what she wants to find out. The experiences she goes through are strange, sometimes outright bizarre, and her world comes crashing down on us. The Handmaids Tale is very thought-provoking, the future of women and indeed the world lies in the actions of todays society, and Atwood uses her perceptions of the present world to support the background of her novel. Altogether The Handmaids Tale offers what all novels should: love, loss, action, comedy (ironic, but appropriate) vision, and plot. It plays with all emotions. Time In The handmaids tale (THT) the use of time is a key feature. Frequently throughout the book we experience time changes, from the present oppressive situation, and to the past of the handmaids, a happier time. In the gymnasium, time is used in reference. The narrator refers to a time gone, where the gymnasium was used for things other than sleeping. Dances would have been held there there was old sex in the room. There is reminiscence of the narrator; they call upon personal observations and experiences from the time gone by I remember that yearning. Later in the first chapter it becomes clear that the narrator, experienced the handmaids experience when she remembers how things were for her, when she slept in the army cots in the gymnasium as we tried to sleep in the army cots she uses words such as we had, then, were which all indicate its past tense. This usage of time goes on in the novel and is a way in which the writer can convey the feeling that the current situation has not always been that way, and that once this oppression didnt exist. As you read the opening chapter, the tone of the text comes across as sad, as reminiscent, as a longing for the times gone by, and a desire to return there. From reading the text, it becomes clear to me that this phantom narrator learned from her experience that she presumably had in Gilead, she learned the talent to be insatiable; she obviously didnt have it when she arrived how did we learn it, that talent for insatiability? The narrator of the text is left as something of a mystery to the reader. A name is never mentioned, but the text reads as if its somebody who is thinking back on their personal experience. This is somebody who has been there, experienced the oppression, had a yearning, this tells me that its being told by somebody who once was a Handmaid, I am sure they were a handmaid as they were being watched by Aunts and sleeping en mass in Gilead. If this person was a handmaid then surely they were a woman, I also think this because the language is quite emotional and emotive we yearned for the future.

The Effects of Global Warming On Coral Reefs Essay -- Environment Glob

The Effects of Global Warming On Coral Reefs Graphs Missing Introduction: The effects of global warming touch every human, animal, plant, ocean, landmass, and atmosphere level on this planet. The numerous effects of global warming are mixes of "good" and "bad" results, depending on how your definition of "good" results and "bad" results are. A "good" effect, a person could say, would be for regions with normally cold temperatures to receive warmer temperatures for their normal. Yet, there are more "bad" effects that seem to out weight the "good" effects. Some of the effects would include increases of flooding, severe storm systems, and rising sea-levels. One major consequence would be an increase of temperature globally. This would give a chain reaction that would change temperatures and precipitation within many ecosystems. Which could cause a possible alteration in migration routes of various animals or produce permanent damage to creatures and their habitats, or worse, result in extinction for sensitive organisms that cannot handle the change. An exampl e of a sensitive organism is the coral reef. This vital creature serves as a home, feeding area, and shelter for many fish, plants, and animals living in the shallow water domain. The degradation by global warming of this essential species is discussed more in-depth below. Bleaching of Coral Reefs When coral reefs are thought of, warm images of vibrant multi-colored creatures and corals emerge from our imaginations. Mental pictures of a bustling biodiversity of animals, invertebrates, and plants congregate around the coral reef that acts as a glue holding together the shallow waters of the underwater realm. Yet, many of the worlds most beautiful and important coral re... ...blic about the bleaching of corals. Its our problem, we need to create a solution or the colors of the ocean may be fading away as we speak. Works Cited Australia Institute of Marine Science (AIMS). "What is coral bleaching?" 2002. http://www.aims.gov.au/pages/research/coral-bleaching/coral-bleaching.html (3 Feb. 2003). Dennis, Carina. "Reef under threat from 'bleaching' outbreak." Nature, 415. 28 February 2002: 947. January 25, 2003. http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v415/n6875/full/ 415947a_fs.html. Houghton, John. 1994. Global Warming The Complete Briefing. Elgin, Illinois: Lion Publishing. Wilkinson, Clive. "The 1997-1998 mass bleaching event around the world." Status of coral reefs of the world: 1998. 1998: Chapter 1. February 4, 2003. http://www.aims.gov.au/pages/research/coral-bleaching/scr1998/scr-00.html.

Justification for Torture

Torture is a scenario that dates back to the old government ages. Is government torture ever justified? This answer is can be answered by this quote,†Torturing the terrorist is unconstitutional? Probably. But millions of lives surely outweigh constitutionality â€Å"(Levin 1). Who would not save the lives of fellow citizens If the only option to solve the issue was torturing an individual for his crimes? Many people would see the situation as a sacrifice for the better of the people. The lives of Americans should be the most important priority of a nation. Torture is justified as long it is for the safety of the people and not for the mere cause of violence. Torture is the considered to be a harmful mentally and physically to anyone. It is a very well known method in the time of war. There have been many cases where it used on children and women. The course of torture can range from physical abuse to sexual abuse. Those are without a doubt an international crime. Using torture on innocent human being is completely out of the question. The use of torture in terms of water boarding on terrorist can be another matter. Torture without a doubt is a cruel treatment for an individual but it must be a nation policy (Falbaum 15). Saving innocent lives should be a governments priority in a â€Å"ticking time bomb † scenario. The scenario states, lets say an officer, got control of a terrorist planning an attack on hundreds of people. It is highly likely this terrorist will be interrogated by other officials to obtain information about the attack. Torture should be an option if the individual refuses to give the data to save hundreds of lives. Falbaum also mentions that 9/11 attack could been prevented if the United States had torture as part of their system. If the US policy was more strict at those times the terrorist would have feared their policy or would have captured their leader and gained details about the whereabouts of the attack. A poll shows that majority of registered citizens prefer harsh techniques that can keep America safe (Bauer 27). The society should support inhumane behaviors like torture if it is seen as last hope for innocent lives. Successful interrogations can keep America safe for the future. The world revolves around information and if your country is lacking information, terrorists can take advantage of the situation. Using torture on terrorists to gain information will benefit in prevention of future attacks on the country. Marc A. Thiessen goes into detail how many terrorists that have been interrogated provided them with reliable information that ended up saving thousands of American lives. â€Å"Thiessen believes that terrorist suspects will talk if pressured in the right way† (Thiessen 36). Some people may argue that torture does not provide the right detail because the victim will say anything to stop the deadly act. A statement said by the terrorist Abu Zubayah explains that captive terrorist tends to resist as long as they can, if they fail to do this they are given permission by their religious belief to tell what they have known. â€Å"The job of the interrogator is to safely help the terrorist do his duty to Allah, so he then feels liberated to speak freely â€Å"(Thiessen 40). It leads to show if the US practice and enhance their torture techniques it will lead to a safer nation. What techniques can be considered torture? Water-boarding is the act where the victim is drained water on his head to stimulate the pain of drowning. Water-boarding to Joseph Farah is not considered torture. He states that the technique is relatively safe with the right management. Not only that he argues that United State also practice this technique on their own soldiers as training (Farah 82). According to him while water-boarding is very abhorrent it is not as bad as cutting of a hand or anything gore. The truth is Water-boarding is very effective and most of the time the result are given in less than a minute. The success stories range from learning about al-Qaida's top plans. It is said that the terrorist agents resisted normal interrogations for hours and hours but gave into water-boarding sooner than the word can be said. This scenario can happen anywhere in the nation. Lets say an al-Qaida member was captured by the CIA and found whereabouts of bombs being placed around a city. They hire the top of the line negotiators to spoil the information. The real question still remains. Will they be able to make the terrorist member talk before it's too late? Then again we could resort to the use of water-boarding and save their lives before it is too late. It is not like we have torn their limbs off to order to get the information. â€Å"But a few seconds of dripping water on a prisoner's face? That's not torture to me â€Å"(Feah 85). Some believe that torture disregard true American standards. Patrick J. Buchanan gives a great insight about this issue. He states that torture is viable as long as it fit to save the American people. It is just like how murder is seen unethical yet you see soldiers and cops given the permission to kill as long as it fits the right scenario (Buchanan 55). Would a war hero that killed twenty men in order to rescue 2,000 citizens receive a death sentence? Just because he murdered 20 men, no. He would be honored as a hero he so deserved to be. There are doctors that inflicts pain to many soldiers in the war field by cutting of parts of their body. This saves their life by preventing death by decay. These are one of the exceptions that torture should be given into. Take for example someone's son or daughter was captivated by a terrorist group. One of their remember was found, he would not spoil or hint any information by simple conversations. What will a parent would want from the government to do? It does not matter if they oppose torture or not, a parent would choose water-boarding compared to the death of their child. Torture is not to be taken lightly but should be considered an option if the situation arise. Torture should the last option if it ever to be used. It a very sensitive topic for many people. There are groups that strongly rejects torture in any situation and there are people that support torture if it leads to saving lives. Ultimately saving lives should be the priority of any government. If it result in agreeing to an â€Å"harsh techniques † like water-boarding to stop future 9/11 attacks, the majority will agree. It is very hard to come to a conclusion about this specific topic because both sides can provide very good viewpoints. It will be one of those controversial topic that will go on for decades.

How Does Golding Show Conflict in Lord of the Flies? Essay

In lord of the flies, Golding presents a strong sense of conflict, between the boys, the boys and nature and the boy’s personal lives. The theme of conflict is an important one within the novel, as it helps to represent the disagreement and fighting of World War II which was happening during the time that Lord of the flies was written and that Golding experienced first-hand. Golding’s main message in Lord of the flies was how cruel men could be to one another and conflict is an obvious link to this idea. We begin to see conflict very early in the novel, even before the story begins, as Golding tries to introduce the key themes at the start of the novel. The boys are actually brought to the island itself by the conflicts of World War II. While conflict and violence is happening in the macrocosm that is the world, it starts to arise on the island, a microcosm of the world. We also see conflict and tension between Ralph and Piggy for a number of reasons. Ralph straightaway thinks of himself as superior to Piggy because he is physically more attractive and athletic then Piggy. While Jack is described as a ‘fair boy’ a stereotypical sign of goodness and pureness who ‘might make a boxer as far as width and heaviness of shoulders went’ and with ‘size and attractive appearance’ that ‘marked him out’. Piggy on the other hand is ‘plump,’ ‘shorter than the fair boy and very fat’. Ralph also mocks piggy by calling him by the nickname he dislikes. This is a deliberate act of cruelty as Piggy says confidently: ‘I don’t care what they call me..as long as they don’t call me what they used to call me at school†¦They used to call me â€Å"Piggy†.’ But Ralph ignores this and mocks Piggy, he ‘shrieked with laughter †¦Jumped up..†Piggy!†Ã¢â‚¬â„¢. He is also rude to him by disregarding his health problem, saying, ‘sucks to your ass-mar!’. This immediate superiority Ralph feels over Piggy and his cruelty towards him could be a representation of how people in society generally act and think, and the behaviour towards thinkers or people with disabilities, that appearance and physical attractiveness is important and superior and that health impairments such as very poor eyesight and asthma marks so meone out as being different and therefore strange. It is also important that Golding has shown that Ralph has the capability to be cruel. Although we later find out that he is a representation of democracy, Ralph as a person can’t be holey good but he can’t be holey evil either. This links to Golding’s main message of Lord of the flies, which was inspired by seeing the awful things men could do to one another: that man, no matter how good, democratic or orderly can be holey good and are capable of being cruel, mostly for no valid reason and often as a result of discrimination and hunger for power. We also see a feature of emotional conflict between the boys. Ralph’s father is in the navy, another person who contributes to the conflict within the world and the island. As well as this, he is not only in the navy but a commander in the navy and taught Ralph how to swim. When Ralph asks Piggy about his father, piggy ‘flushes suddenly’ and replies: ‘‘My dad’s dead’ he said quickly, ‘and my mum—‘†¦Ã¢â‚¬â„¢I used to live with my auntie†¦Ã¢â‚¬â„¢. His words give us the indication that his father is dead, his mother has left him and that his auntie used to look after him but no longer w ants him. His upbringing, especially in comparison to Ralph’s, makes him feel insignificant and upset and evokes pathos in the reader. From this, and his cockney accent (‘all them other kids’, ‘we was attacked’) we can conclude that Piggy represents the underclass in the 1950s. However, out of all the boys, the majority of which attended private school or were educated in grammar school, piggy is the best one, even though he has had to live in care and has been neglected through his childhood, all things which Golding uses to make the reader feel empathy and sympathy towards him. As a character, Piggy is the cleverest among the boys: ‘what intelligence had been shown was traceable to piggy’ but he is really a weak boy with good ideas. Golding could be making a point that your background does not necessarily determine how intelligent you are and that a lower class child can be very intelligent. We experience more tension between Ralph and Piggy due to Ralph’s attitude that he is superior, through linguistic conflict. Piggy continuously asks Ralph questions, such as ‘what’s your name?’, ‘you haven’t seen the others have you?’ and making comments and telling Ralph about himself: ‘My auntie told me not to run†¦on account of my asthma’ and ‘I expect we’ll want to know all there names’. He is enthusiastic and keen to talk to Ralph, possibly due to the usual lack of attention he receives back home. However Ralph ‘tried to be offhand and not too obviously uninterested’ and replied to Piggy’s comments with just a shake of his head or not even acknowledging him at all. As well as this, when Piggy asked Ralph his name he ‘waited to be asked his name in turn but this proffer of acquaintance was not made’. This yet again shows how Ralph thinks of himself as superior to Piggy and doesn’t feel he needs to treat or show piggy the same respect and interest that he gave to Ralph. Piggy also partly creates a gap between himself and Ralph by following Ralph and showing so much interest in him. Piggy ‘hung steadily at his shoulder’ and ‘stood by him, breathing hard’. This could show that Ralph has a natural sense of leadership about him which makes Piggy feel as though he should look up to him. However, it is mainly due to the fact that Piggy is neglected, of love and also friendship, which makes him feel he needs to attach himself to someone like Ralph. As well as this, Piggy is frightened by the prospect of ‘no grown-ups’ and needs a sense of authority, like Ralph, to latch onto. We can see how nervous and lacking in confidence Piggy can be, due to neglect as he ‘took off his glasses†¦then started to wipe them’, an action he seems to be doing all the time. This fidgeting behaviour is especially seen when Piggy is feeling particularly insignificant and upset talking about his upbringing: ‘†My dad’s dead,’ he said quickly, ‘and my mum—‘†¦. He took of his glasses and looked vainly for something with which to clean them on’. Another important conflict within the book can be seen between two of the main leaders, Ralph and Jack. Before the boys even interact with each other we can foresee that there will be tension between the two characters. Jack’s choir is describes a ‘creature’. This description could indicate that the choir (le d by Jack) cloud later become more savage, like a ‘creature’. As well as this, the colour black that Jack and the choir’s uniforms are made from is symbolic, with black being symbolising evil and bad things. The description of Ralph is set in antithesis of Jack – Jack is described as ‘black’ and ‘ugly’ and Ralph as ‘golden’. Later in the book we see that the two boys makes choices and live in ways that would be expected by their description, Jack as savage and cruel and Ralph as democratic and reasonable. Golding could be implying that one can make valid judgements from appearance. He also writes that Jack was ‘underneath the floating cloak†¦tall, thin and bony. His face was crumpled and freckled and ugly with silliness’. This description once again draws up an image of Jack being a ‘bad’ character because of his ugliness. However, by using the modifier ‘without t silliness’, Golding modifies the statement on Jack’s appearance, that although he is ugly, the children would not be able to laugh at him. This could represent Jack’s power and intimidation towards the other children through sheer appearance and manner. Jack also speaks in imperative sentences (e.g. ‘choir, stand still! ) and the boys ‘huddled together’ in fear of him and stand with ‘wearily obedience’. This shows us they are used to taking orders from Jack. We can also see that Jakc thinks of himself as having superiority, as he wants to be called by his surname ‘Merridew’. Jack is obviously from a posh background and will have been to private school. Having characters like Jack and Ralph as leaders, boys who attended public school, could be Golding’s representation of society, how many leading roles and responsibilities are taken by upper-class privately educated people, while many lower-class people, some who may be intelligent like piggy, are left behind, as they have not been in an environment where being confident and superior is normal and expected. When it comes to actually voting for a chief, ‘the most obvious leader was Jack’, described by Golding as ‘this was the voice of one who knew his own mind’. This shows us that Jack already asserts himself as a leader in opposition to that of Ralph who doesn’t. However it is Ralph who is chosen by the boys to be chief. It is not only ‘his size and attractive appearance’, but there was also ‘a stillness about Ralph†¦that marked him out’ and ‘most powerfully there was the conch†¦the being that had blown that†¦was set apart’. The conch links to democracy, order and civilised society and there is a link between Ralph, ‘the being that had blown that’, that the boys also see. Golding would have been making the point that the boys chose, even when they didn’t know him, Ralph, the link to democracy, order and civilised society, to be their chief because a democratic leader is the right leader to have and the boys can see this and therefore choose Ralph even without knowing him. When Jack did not get voted as chief, ‘the freckles disappeared on jack’s face under a blush of mortification’. This shows us how embarrassed, angry and upset Jack was for not being chosen as Chief. This could be a point of conflict between Ralph and Jack within the book, however Ralph tries to keep peace with jack and ‘looked at him, eager to offer something’. This demonstrates Ralph’s eagerness to be a good and fair leader. However, another point at which Ralph tries to avoid conflict between Jack and himself is when laughs at Jack’s name calling at piggy and says, ‘he’s not fatty†¦his real name’s Piggy!’. This brings Ralph and Jack closer and creates common ground between them but yet again demonstrates the fact that Ralph can be capable of cruelty. We also see Ralph’s attempts to avoid conflict with Jack over the role of leader by allowing jack that ‘the choir belongs to you obviously.’ This action gives Jack some leadership and makes him feel more powerful as well as foreshadowing Ralph’s later attempts to break the conflict between him and Jack and bring the two together again. Nearing the end of the chapter, we experience the boy’s conflict with nature. This is represented by Golding, when the boys go to push a rock down from its original place on the mountain top. During their attempt, ‘the great rock loitered, poised on one toe’. This behaviour, and other actions, is typical of the way humans have often treated the planet, destroying natural objects or areas from their original state for human need and want. Golding was aware that humankind is stupid enough to destroy the very land that gives it food and life and we see this idea explored further in the novel, when the boyâ₠¬â„¢s destroy a lot of food and firewood, elements that keep them alive, in a n uncontrollable fire. This, and the rolling of the rock, demonstrates that even if the boys are intelligent or strong humans, they will never really overpower nature and that it will always be in conflict with them. As well as this, the rolling of the rock down the mountain side is proleptic of Piggy’s death. We later also see that Jack experiences self-conflict. During Simon, Ralph and Jack’s expedition of the island they come across a pig which Jack tries to kill, but isn’t able to. He felt he couldn’t, as he understood ‘what an enormity the downward stroke would be’. The boys also ‘knew very well why he hadn’t: because of the enormity of the knife: descending and cutting into living flesh: because of the unbearable blood.’ This shows us that Jack is not yet be far enough removed from civilised society to be able to kill a pig. However, he ‘snatched his knife’ and ‘slammed it into a tree trunk’, saying, ‘Next time!’ and ‘he looked round fiercely, daring them to contradict’. This shows that his natural, evil, menacing and savage instincts are in him that only now on the island are recently coming across. Notice how Golding uses the verb ‘flesh’, a word which is not clearly specific to a pig and could easily be confused with the ‘flesh’ of a human. This shows us that when jack does ‘next time’ kill, his knife could be coming down into the flesh of a pig, or the flesh of a human. It is important that Golding has introduced the variety of conflicts in the first chapter, so that the key themes are established early on and can develop throughout the book and as to foreshadow events that will happen later on in the novel.

The Truth About DNA Fingerprinting essays

The Truth About DNA Fingerprinting essays Mr. Doe is 52 years old. He has lived in the countryside of Minnesota his entire life. He has no serious diseases and is living a normal life. One day, he receives a letter from his life insurance provider making him aware that his annual life insurance premium has increased 12 percent. Mr. Doe proceeds to call his insurance provider, and asks why the sudden and abrupt increase in his premium. The life insurance provider tells him that he will probably develop Alzheimer's by the age of 57. Mr. Doe is awkwardly confused. He has no idea how the life insurance provider knows he will probably get Alzheimer's later in his life. The life insurance provider tells Mr. Doe that according to his genetic sequence, he is likely to suffer from Alzheimer's. Although this scenario may seem a bit far-fetched, with today's technological advances, scientists are able to outline the genetic makeup of humans, allowing them to predict what diseases one may be prone to. Now, you might be thinking "This is great, now I will be able to protect myself from diseases, right?" Maybe so, but will you be able to protect your assets and personal information from major conglomerate corporations trying to make a profit from your foreseen agony? Along with this situation, many other issues arise concerning the knowledge of your personal genetic information. DNA, or deoxyribonucleic acid, is found within every cell of living things. The function of DNA is to carry and store genetic information for the cell. DNA is made up of two chains composed of deoxyribose sugars and phosphates that form a double helix twist. Each deoxyribose sugar is covalently bonded to the phosphates. The deoxyribose sugars are also covalently bonded to nitrogenous bases. These nitrogenous bases, known as adenine, thymine, guanine, and cytosine, are bonded together with hydrogen bonds. Adenine bonds with thymine and cytosine bonds with guanine. According to this base pairing rule, everyone has ...

How to Become a Dietitian or Nutritionist

How to Become a Dietitian or Nutritionist As Americans face a growing obesity crisis and ever-growing awareness about how what we eat affects our health, medical professionals in the field of dietetics are key members of that front line. Behind every public program like healthier school lunch initiatives or campaigns to fight Type 2 diabetes, dietitians and nutritionists are the ones using science to set healthy food guidelines and diet plans. The Day-to-DayDietitians and nutritionists work in a variety of settings, from healthcare settings (hospitals and clinics) to government (public health agencies) to the private sector (food manufacturing and distribution companies). They may work directly with patients to create and maintain diet plans to lose weight or improve health, but they might also work on larger-scale public health programs to encourage healthy eating to broader populations. You can also find dietitians working on nutrition guidelines and food safety in the government or in private companies. Their tasks often include:Designing diets that target specific conditions, like obesity, diabetes, or high blood pressureHelping patients maintain diets for health or weight loss, and ensuring that patients are dieting safelyDeveloping nutrition programs for an entire facilityImproving accuracy in food labels and advertisingWorking with agencies and manufacturers to improve food safetyResearching how food and nutrition interact with the body and various conditionsEducating the public (broadly or in specific targeted populations) on nutrition, food safety, and healthy lifestyle practicesDietitians and nutritionists typically work a standard full-time work week in an office or clinic setting.  For more on dietetics and what it’s like to be a dietitian or nutritionist, check out these videos:A Day in the Life: DietitianHow to Become a Registered Dietitian/NutritionistAsk a Nutritionist with Shira LenchewskiIs Being a Nutritionist Right for You?The RequirementsMost dietitians and nutritionists h ave a bachelor’s degree, and have completed a residency, internship, or other form of supervised training. Most states require a license for dietitians and nutritionists, so be sure to check your own state’s requirements if you’d like to start down this career path.The SkillsThe dietetics field calls for a number of special skills and knowledge bases, including:Attention to detailMath and science (particularly biology, food science, and biochemistry)Critical thinkingPatient evaluationAnatomy and physiologyDisease managementPublic health implementationThe PayPer the U.S. Bureau of Labor Statistics (BLS), the median salary for dietitians/nutritionists is $56,950, or $27.38 per hour. In addition, the field offers a lot of fulfillment for its members. According to a survey conducted by PayScale, the average dietitian is â€Å"extremely satisfied† in his or her career.The OutlookAs mentioned before, the renewed focus on food and nutrition, both on a national a nd personal health level, means that this is a gangbusters-level field for growth. Openings in dietetics are expected to grow at least 16% by 2014, which is significantly faster than average.Interested? APPLY HERE

Organisational Behaviour College Essay Example | Topics and Well Written Essays - 1500 words

Organisational Behaviour College - Essay Example We can analyze Nucor by Robbins (2001), reinforcement theory ignores the inner state of the individual and concentrates solely on what happens to a person when he or she takes some action. Significant research indicates people will exert more effort on tasks that are reinforced than on tasks that are not this statement is definitely proven by Nucor as pay day is always a time to celebrate for the employees in 2005 they distributed $220 million and this made the employees work even more hard for them. Thus as Robison pointed they exert more efforts in their takes. Nucor realizes this shares its profit with its employees so that they can get the maximum from them Reinforcement theory will work well for Nucor's employees because they thrive on individual recognition and with little time and effort will become comfortable with being recognized as part of a team. Reinforcement theory works well for Nucor because employees are likely to put forth more effort if they know that same effort will be rewarded when the task is finally completed. The reward itself is not as important as knowing that there will be a reward. Q2).What Role Does Equity Theory Play In The Case Let us starts by explaining exactly what this theory is all about the equity theory states: employees weigh what they put into a job situation (input) against what they get from it (outcome) and then compare their input-outcome ratio with the input-outcome ratio of relevant others (Robbins, 2001 p115). It can also be said that Equity theory gives complete attention to on the feelings of employees of how fairly they have been treated in contrast with the treatment which other employees get " (Laurie 2007 p.435). Equity does play a huge role in the case , as Nucor knows that Equity plays an important role for employees. If an employee feels equally treated, he or she will sense fairness. If an employee feels they are treated unfairly, they may feel they are not being treated well enough. Thus when the company does not make enough profit or a bad batch of steel goes into market every one looses out on the bonus and profit sharing. By everyone I even mean the CEO and top management what more equity could the employee ask for Since each employee is an individual, equality is an important aspect of maintaining an effective. When there are individual differences among employees, there are also potential workgroup conflicts but Nucor solves this problem by making sure that bonuses are calculated every week so that every employee gets a bit of the cash. Robbins (2001) states that there are five different choices an individual might make if faced with inequity. They are as follows: distort either their own or others' inputs or outcomes, behave so as to induce others to change their inputs or outcomes, behave so as to change their own inputs or outcomes, and/or choose a different comparison referent or quit their job (p.

Introduction to E-Commerce Coursework Example | Topics and Well Written Essays - 2000 words - 2

Introduction to E-Commerce - Coursework Example The case of â€Å"The Royal Automobile Club† and â€Å"Cabela’s† will be discussed in this report. These are two businesses which expanded their operations through e-commerce. Enhancement of the web portals increased the sustainability of the businesses. The new platform anchored by the businesses provided the customers with a world-class shopping experience. This enhanced the domain of retail selling of the businesses at a whole new level. This report will emphasize on assessing the findings and analyzing the benefits which the businesses have availed through this practice. The findings in this section have been supported by applying two cases which have elaborately been placed in the appendix. The first case is of Cabela’s which is a retail store business which supplies fishing, hunting, and camping equipment. Cabela’s has several physical stores located and it is engaging with e-commerce for providing ease of accessibility to different customers and at the same time enhance the existing customer base (Micros, 2013). Similarly, the other case of the Royal Automobile Club shows that to facilitate its existing members the business has initiated entering in the e-commerce business. This gives the club members flexibility of reserving the restaurant and rooms at the club in the easiest possible manner (Micros, 2012). Both the cases are adopting e-commerce but the purpose and customers.

Japanese contemporary art and Korean contemporary art Coursework

Japanese contemporary art and Korean contemporary art - Coursework Example The essay "Japanese contemporary art and Korean contemporary art" presents contemporary art practice in Japan and Korean, comparing and contrasting their different attributes. The paper also attempts to highlight the diverse effects that art have on the respective communities. The introduction of religion had a great significance to the Korean artistic images, and artistic styles were manifest in the temple developments, bronze statues, portraits and exemplified manuscripts. Also, Korea’s physical position at the crossroad of East Asia- between its two larger neighbors, Japan, and China – added an enormous influence on its history and culture and its artistic fundamentals. Korea functioned as an outlet between China and Japan for philosophies and theories and technologies that enriched Koreans artistic innovation and skills. Moreover, scholars have established non-passive role of Korea of spreading artistic ideas across to Japan or China, and recognized it not only diff used culture but also integrated it resulting to unique Korean art and culture of its own. Japanese art covers broad assortments of art styles and media that include ancient poetry, wood and bronze sculptures, silk and paper ink paintings, and other type artwork. Japanese painting has a rich history of synthesis and rivalry between natural Japanese aesthetics and utilization smuggled ideas. Further Japanese sculptures mainly originated from the idol reverence in Buddhism or animistic rites of Shinto deity. Particularly.

Impact of Foreign Banks on Banking in Emerging Economies Essay Example for Free

Impact of Foreign Banks on Banking in Emerging Economies Essay Increased technology and innovation International banking in emerging–market have some advantages from the technology and innovation. The advanced technology and innovation system could even surpass the conventional technology and innovation. For example, they could improve productivity, increase in market and increase the competition and so on . Innovations in customer experience and superior customer service delivery, network integration. (Infosys 2000). For example, the internet and computer system have a useful communication system to connect the consumer and bank. In daily life, customer often use the mobile phone, computer transfer the money. At the same time, innovation and technology is a lower cost of the banking system in the emerging market. The increased technology and innovation in emerging market may help the banking system make a clear communication for their employee, shareholder and consumer. As a result, banks in emerging markets are leapfrogging their rich-world rivals in efficiency, technology and innovation (special report international banking 2011). Increased liquidity and solvency Comparing with the local banking system, the foreign banks on banking in emerging market have different kind of comparative advantage. The reason is emerging market allow foreign bank entry to local market. This is lead to the higher liquidity and solvency. Foreign direct investment is a useful fund source for local market. At the same time, the foreign banks also have important roles which represent a borrower. For example, foreign banks have an enough capital base and asset. Foreign banks have played a major role in financing emerging market (EMEs) in recent year. Increased liquidity and solvency has helped emerging markets to develop their economies and allocate capital and financial know-how efficiently across countries (Agustà ­n Villar ) Disadvantage Complex global policies and challenges international banking There are some negative factors occur in global banks in emerging economies. One of the important factors is complex global policies. For example, the foreign banks are an extension of parent bank which sent to managers to overseas. Different banking system has different policies. Meanwhile, the government also comes up with stricter policies. As a result, foreign banks should face a lot of complicated policies in emerging market. The collapse of Barings was a demonstration of how different countries supervisors are failing to communicate with each other.( the economist 1997). This opinion shows that the international banking in emerging market should have a closer supervisor.

The Fight for Human Rights Essay -- the security-for-rights compromise

Can you imagine a life without pre-meditated murder? In his movie Minority Report, Steven Spielberg brings this vision to reality in the trappings of a police state. The pre-crime unit is charged with the elimination of pre-meditated murder using three pre-cogs, humans with the ability to predict violent crime. Minority reports- sporadic, erroneous predictions- indicate the fallibility of this system of imperfect procedural justice. Civilians have their rights to privacy violated on a regular basis for collection of intelligence. This movie is chillingly pertinent in the real world, as today African-Americans and Muslim-Americans have their rights violated regularly in the name of security. Thankfully, we have more than a Hollywood protagonist to fight for the protection of rights. John Stuart Mill, Robert Nozick, and John Rawls provide a philosophic framework for evaluating the security-for-rights compromise. Though their respective theories vary greatly in theory and in practice, they provide models to condemn this exchange. Nevertheless, each differs in the persuasiveness and effectiveness of their tools for argumentation. Mill’s utilitarianism, Nozick’s libertarianism, and Rawls’ egalitarian liberalism reject the tradeoff of security for a majority in exchange for the violation of the rights a minority. John Stuart Mill outlines a sometimes dubious plan for protecting rights and lacks the a priori protection of rights that Rawls and Nozick afford. John Rawls presents the most convincing and solid argument for the omnipotence of rights in the confines of a welfare state. His philosophy acts as an ideal synthesis of libertarianism and utilitarian ism; he demands the utmost respect for rights while trying to maximize utility f... ...t for rights because Tom Cruise says so and because Mill, Nozick and Rawls prescribe it. Works Cited Bentham, Jeremy. Introduction to the Principles of Morals and Legislation. Published online by Constitution Society. Web 18 June 2015. http://www.econlib.org/library/Bentham/bnthPML.html Mill, John Stuart. "Utilitarianism." Web 20 June 2015. http://www.utilitarianism.com/jsmill.htm Nozick, Robert. â€Å"Distributive Justice.† Macquarie University, Modern Political Theory. Web 18 June 2015. http://www.csudh.edu/dearhabermas/nozick01bk.html Rachels, James. The Elements of Moral Philosophy. Birmingham, Alabama: Mcgraw-Hill College, 1999. Rawls, John. A Theory of Justice. Harvard, MA: Harvard University Press, 1999. Minority Report. Dir. Steven Spielberg. Perf. Tom Cruise, Max Von Sydow, Steve Harris. Videocassette. 20th Century Fox, 2002.

Project Charter for Payroll System Essay

1Project Background 1.1Problem/Opportunity Description The following problems or opportunities listed below are organizational problems that the proponents discovered after analyzing the process of not having a system for employee’s payroll: Manual Payroll can’t handle large numbers of employees; Multiple works and positions of an employee makes the payroll more complicated; Searching of data (Data Mining) when particular data is needed due to an urgent matter can be very hard for both employees and owner of the business firm; Difficulty in managing and recording of information on work schedule, hours worked units of pay, deductions and leave of absences, distribution of exact amount of wages and salaries in manual payroll that causes delay and sometimes insufficiency in amount of compensation given to employees during payday; And lastly, not all Payroll System are generic. 1.2Benefits One of the most important tasks in running a business is completing payroll. The following shows the prospect hospital benefits in having Payroll System and proponents benefits as well: A fair, on-time, and consistent distribution of income or organization; Accurate recording of time and attendance, information on work schedule, hours worked, units of pay, deductions and leave of absences in the system; Manageability in employee’s work units so that exact amount of wages, salaries, and other bonuses will be financially given on payday; Calculate benefits, taxes and dependencies of the employee; Payroll System that can be used by any hospitals; 1.3Goals Goals are the purpose and direction to the project. The proponent’s goals in Payroll System are the following: Create a system that can cater large number of employees’ payroll; A system that may allow the possibility of having employee’s multiple work units; A system that can accurately record time and attendance, information on work schedule, hours worked, units of pay, deductions and leave of absences in the system; A system that can ensure the exact amount of wages, salaries and bonuses that will be given to employees on payday; A system that is generic and can be used to any company or establishment’s payroll; And lastly, to provide a Payroll System that will be as functional as the other Payroll Systems. 1.4Stakeholders and Clients The stakeholders and clients who will be involved and support this project are as follows: Companies who do not have and would like to have a system for the payroll The proponents who will develop the payroll system The project adviser who requires the proponents to have the payroll system as their project The Project Evaluation Committee (PEC) who will give advices and guide the proponents before the development of the project 2Project Scope 2.1Objectives The objectives of the development team on creating the payroll system are the following: To create a system that can manage data of employees and have a solution in creating their salaries without interrupting the process of the other systems; A system that can provide a flexible system that can easily modify a response to any altered circumstances or conditions; A system that can ensure the safety of data from other possible circumstances that may result to corruption and loss of data; A system that have a user-friendly interfaced so that the user who will administer and use the system will not have a hard time; A system that can generate analytical reports at any time; A system that can calculate accurately the benefits, taxes and dependencies of employees; A system that is generic and can be applied or used to any hospitals; A system that attain the same quality and functionality with other payroll system; And lastly, a system that is accessible and well-integrated to other hospital modules. 2.2Deliverables A deliverable is any tangible, measurable outcome of a project. The following tables consists specific end results, products, or outputs of the project for each objectives: Objective 1 – To provide a system that can manage data of employees and have a solution in creating their salaries without interrupting the process of the other systems. Project Deliverable Work Products/Description Manage data of employees Create a system that will ensure the management of employee’s data Ensure that the computation of salaries will be accurate and will not affect or interrupt other related processes in the system Create a database and specialized table that will generate an accurate computation of employee’s salaries Objective 2 – A system that can provide a flexible system that can easily modify a response to any altered circumstances or conditions. Project Deliverable Work Products/Description Provide a system that can modify a response to any possible circumstances, conditions, or situations that may occur in future Consider situations and possibilities to the risks that may be encountered on payroll and create a risk management solution. Objective 3 – A system that can ensure the safety of data from other possible circumstances that may result to corruption and loss of data. Project Deliverable Work Products/Description Ensure data security Provide a secured database. Back-up and archive all the transactions and reports every day. Provide trusted and effective anti-virus software to avoid corruption or immediate loss of files. Objective 4 – A system that have a user-friendly interfaced so that the user who will administer and use the system will not have a hard time Project Deliverable Work Products/Description Create a user-friendly interfaced system Apply the standards and guidelines in choosing template for the system. Use a user-friendly template for system’s interfaced so that anyone who will use it will not have a hard time. Objective 5 – A system that can generate analytical reports at any time Project Deliverable Work Products/Description Generate analytical reports Include analytical reports that can be generated anytime. Objective 6 – A system that can calculate accurately the benefits, taxes and dependencies of employees Project Deliverable Work Products/Description Calculate benefits, taxes and dependencies of employees Ensure the correct amount of benefits, taxes and dependencies Review every deductions that may happen to the salary of employees Objective 7 – A system that is generic and can be applied or used to any hospitals Project Deliverable Work Products/Description Generic Payroll System Know the different payroll processes in different type of company/establishment and apply the rules in developing the system. Objective 8 – A system that attain the same quality and functionality with other payroll system Project Deliverable Work Products/Description Same quality and functional payroll system Conduct a research about payroll system. Schedule and make an interview with hospitals who has a payroll system. Analyze and combine all the information gathered about the payroll system and create a guidelines that will be used system‘s development. Objective 9 – A system that is accessible and well-integrated to other hospital modules Project Deliverable Work Products/Description Follow database standards Follow the standards implemented to avoid problems in integration and to attain the expected output for the system. Create a flexible payroll system that can be integrated to other related systems Ensure that the processes were correct so there will be no problem when integration was applied. 2.3Out of Scope The items listed below may be related but will not be managed as part of the project. This critical important section of project, allows the proponents to defend scope throughout the course of the work, by declining requests to work on items that are clearly defined as out of scope. Items are as follows: Employee trainings Professional Fee 3Project Plan 3.1Approach and Methodology The methods and approaches of the development team to finish the project are classified into three: Data Gathering Process: The proponents will conduct an interview from different hospitals and other companies that can help to the system development Study and research (using Internet) are also essential to gather information about the system Development: The proponents will be using Software Development Life Cycle (SDLC) particularly Waterfall method for the development of the system The system will be built from scratch that will include open source software for the front end and proprietary software for the database. Testing: Create a test plan and test cases to follow the expected and actual results of the system. The development team will also interact to other systems integrated on the system. 3.2Project Timeline The table below shows the project timeline of Payroll System: 3.3Success Criteria The project milestones of this project are the things that must be achieved by the developers to meet their goals and these are the following: Payroll System is fully functional Required reports can be generated All the requirement has been followed Payroll system has become more accessible and conforms on the quality assurance standards Payroll System is well integrated to other Hospital modules in Hospital EIS Generic Payroll System 3.4Issues and Policy Implications The proponents do brainstorming and come up to some dependencies that may affect the process of the system. These are: Human Errors – Wrong input on data in the system Unexpected errors that may cause delays in recording of data Unexpected natural disaster that may cause loss of data Unwanted virus that may corrupt the system 3.5Risk Management Plan There are the factors that can affect the outcome of the project including major dependencies on other events or actions. These factors can affect deliverables, success, and completion of the project. The proponents think actions to some factors that may affect the whole process of Payroll System. The likelihood of each risk are indicated in the Probability and Impact on the project and rated as H (high), M (medium), L (low). 3.6Service Transition (Optional) If the project will change or otherwise impact a previously defined ITS service, here are the proponent’s plan for transitioning project deliverables into service operations. Service transition includes activities such as: Have a system support center to fix problems that may occur while using the system If the support center will have any changes in their management, dissemination of information will be immediately sent through e-mail or phone calls. The implementation will occur only after both parties will approve on the changes. The programmer and other staff who is responsible for the system support may be the one to respond for the system’s problem 3.7Option Analysis We can’t say that the success criteria of the proposed system were a success; here are some options that will help if the process of development has been interrupted: If the interview was not enough, the proponents will ask some IT alumni (who already developed Payroll System) to gather more essential information that the proponents missed during the interview; The proponents decided to conduct another interview to other Hospitals; They will ask programmers who are familiar to the system; If the Production cost was not enough to develop the system, they will ask supports from the IT alumni; If the proponents don’t have a machine (laptop) to develop and create the system, they will rent a laptop or else, if they don’t have a choice, a desktop may do. 4Technical Features This section provides a detailed description of technical requirements stated in terms suitable to form the basis for the actual design development and production processes of the project having the qualities specified in the operational characteristics. Hardware Specifications Hardware Specification Processor Min. requirement of 2.6 GHz and recommended requirements is 3.3 GHz or higher Memory Min. requirement of 1 to 2 GB and recommended requirement is 2 GB or higher Hard Drive Min. requirement of 80 to 250 GB and recommended requirement is 500 GB or higher Servers Dedicated to run one or more services Software Specifications Software Specification Use Windows 7 Operating System For the environment oor platform to be used Java Programming Language For the development of the system MsSQL Database For the storage of records 5Project Organization and Staffing The template provided below includes an organization chart, or both, list of the roles, names, and responsibilities of individuals that will be involved in the project. Role Names & Contact information Responsibilities Executive Sponsor Serve as ultimate authority / responsibility for the project Provide strategic direction and guidance Approve changes to scope Identify and secure funding Project Sponsor

The Higher Arithmetic – an Introduction to the Theory of Numbers

This page intentionally left blank Now into its eighth edition and with additional material on primality testing, written by J. H. Davenport, The Higher Arithmetic introduces concepts and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical signi? cance. A companion website (www. cambridge. org/davenport) provides more details of the latest advances and sample code for important algorithms. Reviews of earlier editions: ‘. . . the well-known and charming introduction to number theory . . can be recommended both for independent study and as a reference text for a general mathematical audience. ’ European Maths Society Journal ‘Although this book is not written as a textbook but rather as a work for the general reader, it could certainly be used as a textbook for an undergraduate course in number theory and, in the reviewer’s opinion, is far superior for this purpose to any other book in English. ’ Bulletin of the American Mathematical Society THE HIGHER ARITHMETIC AN INTRODUCTION TO THE THEORY OF NUMBERS Eighth edition H. Davenport M. A. , SC. D. F. R. S. late Rouse Ball Professor of Mathematics in the University of Cambridge and Fellow of Trinity College Editing and additional material by James H. Davenport CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. cambridge. org Information on this title: www. cambridge. org/9780521722360  © The estate of H. Davenport 2008 This publication is in copyright.Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 ISBN-13 978-0-511-45555-1 978-0-521-72236-0 eBook (EBL) paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. CONTENTS Introduction I Factorization and the Primes 1. 2. 3. 4. . 6. 7. 8. 9. 10. The laws of arithmetic Proof by induction Prime numbers The fundamental theorem of arithmetic Consequences of the fundamental theorem Euclid’s algorithm Another proof of the fundamental theorem A property of the H. C. F Factorizing a number The series of primes page viii 1 1 6 8 9 12 16 18 19 22 25 31 31 33 35 37 40 41 42 45 46 II Congruences 1. 2. 3. 4. 5. 6. 7. 8. 9. The congruence notation Linear congruences Fermat’s theorem Euler’s function ? (m) Wilson’s theorem Algebraic congruences Congruences to a prime modulus Congr uences in several unknowns Congruences covering all numbers v vi III Quadratic Residues 1. 2. 3. 4. . 6. Primitive roots Indices Quadratic residues Gauss’s lemma The law of reciprocity The distribution of the quadratic residues Contents 49 49 53 55 58 59 63 68 68 70 72 74 77 78 82 83 86 92 94 99 103 103 104 108 111 114 116 116 117 120 122 124 126 128 131 133 IV Continued Fractions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Introduction The general continued fraction Euler’s rule The convergents to a continued fraction The equation ax ? by = 1 In? nite continued fractions Diophantine approximation Quadratic irrationals Purely periodic continued fractions Lagrange’s theorem Pell’s equation A geometrical interpretation of continued fractionsV Sums of Squares 1. 2. 3. 4. 5. Numbers representable by two squares Primes of the form 4k + 1 Constructions for x and y Representation by four squares Representation by three squares VI Quadratic Forms 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Equivalent forms The discriminant The representation of a number by a form Three examples The reduction of positive de? nite forms The reduced forms The number of representations The class-number Contents VII Some Diophantine Equations 1. Introduction 2. The equation x 2 + y 2 = z 2 3. The equation ax 2 + by 2 = z 2 4. Elliptic equations and curves 5.Elliptic equations modulo primes 6. Fermat’s Last Theorem 7. The equation x 3 + y 3 = z 3 + w 3 8. Further developments vii 137 137 138 140 145 151 154 157 159 165 165 168 173 179 185 188 194 199 200 209 222 225 235 237 VIII Computers and Number Theory 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Testing for primality ‘Random’ number generators Pollard’s factoring methods Factoring and primality via elliptic curves Factoring large numbers The Dif? e–Hellman cryptographic method The RSA cryptographic method Primality testing revisited Exercises Hints Answers Bibliography IndexINTRODUCTION T he higher arithmetic, or the theory of numbers, is concerned with the properties of the natural numbers 1, 2, 3, . . . . These numbers must have exercised human curiosity from a very early period; and in all the records of ancient civilizations there is evidence of some preoccupation with arithmetic over and above the needs of everyday life. But as a systematic and independent science, the higher arithmetic is entirely a creation of modern times, and can be said to date from the discoveries of Fermat (1601–1665).A peculiarity of the higher arithmetic is the great dif? culty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence. ‘It is just this,’ said Gauss, ‘which gives the higher arithmetic that magical charm which has made it the favourite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics. ’ The theory of numbers is generally considered to be the ‘purest’ branch of pure mathematics.It certainly has very few direct applications to other sciences, but it has one feature in common with them, namely the inspiration which it derives from experiment, which takes the form of testing possible general theorems by numerical examples. Such experiment, though necessary in some form to progress in every part of mathematics, has played a greater part in the development of the theory of numbers than elsewhere; for in other branches of mathematics the evidence found in this way is too often fragmentary and misleading.As regards the present book, the author is well aware that it will not be read without effort by those who are not, in some sense at least, mathematicians. But the dif? culty is partly that of the subject itself. It cannot be evaded by using imperfect analogies, or by presenting the proofs in a way viii Introduction ix which may convey the main idea o f the argument, but is inaccurate in detail. The theory of numbers is by its nature the most exact of all the sciences, and demands exactness of thought and exposition from its devotees. The theorems and their proofs are often illustrated by numerical examples.These are generally of a very simple kind, and may be despised by those who enjoy numerical calculation. But the function of these examples is solely to illustrate the general theory, and the question of how arithmetical calculations can most effectively be carried out is beyond the scope of this book. The author is indebted to many friends, and most of all to Professor o Erd? s, Professor Mordell and Professor Rogers, for suggestions and corrections. He is also indebted to Captain Draim for permission to include an account of his algorithm.The material for the ? fth edition was prepared by Professor D. J. Lewis and Dr J. H. Davenport. The problems and answers are based on the suggestions of Professor R. K. Guy. Chapter VIII a nd the associated exercises were written for the sixth edition by Professor J. H. Davenport. For the seventh edition, he updated Chapter VII to mention Wiles’ proof of Fermat’s Last Theorem, and is grateful to Professor J. H. Silverman for his comments. For the eighth edition, many people contributed suggestions, notably Dr J. F. McKee and Dr G. K. Sankaran.Cambridge University Press kindly re-typeset the book for the eighth edition, which has allowed a few corrections and the preparation of an electronic complement: www. cambridge. org/davenport. References to further material in the electronic complement, when known at the time this book went to print, are marked thus:  ¦:0. I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3, . . . of ordinary arithmetic. Examples of such propositions are the fundamental theorem (I. 4)? hat every nat ural number can be factorized into prime numbers in one and only one way, and Lagrange’s theorem (V. 4) that every natural number can be expressed as a sum of four or fewer perfect squares. We are not concerned with numerical calculations, except as illustrative examples, nor are we much concerned with numerical curiosities except where they are relevant to general propositions. We learn arithmetic experimentally in early childhood by playing with objects such as beads or marbles. We ? rst learn addition by combining two sets of objects into a single set, and later we learn multiplication, in the form of repeated addition.Gradually we learn how to calculate with numbers, and we become familiar with the laws of arithmetic: laws which probably carry more conviction to our minds than any other propositions in the whole range of human knowledge. The higher arithmetic is a deductive science, based on the laws of arithmetic which we all know, though we may never have seen them form ulated in general terms. They can be expressed as follows. ? References in this form are to chapters and sections of chapters of this book. 1 2 The Higher Arithmetic Addition.Any two natural numbers a and b have a sum, denoted by a + b, which is itself a natural number. The operation of addition satis? es the two laws: a+b =b+a (commutative law of addition), (associative law of addition), a + (b + c) = (a + b) + c the brackets in the last formula serving to indicate the way in which the operations are carried out. Multiplication. Any two natural numbers a and b have a product, denoted by a ? b or ab, which is itself a natural number. The operation of multiplication satis? es the two laws ab = ba a(bc) = (ab)c (commutative law of multiplication), (associative law of multiplication).There is also a law which involves operations both of addition and of multiplication: a(b + c) = ab + ac (the distributive law). Order. If a and b are any two natural numbers, then either a is equal to b o r a is less than b or b is less than a, and of these three possibilities exactly one must occur. The statement that a is less than b is expressed symbolically by a < b, and when this is the case we also say that b is greater than a, expressed by b > a. The fundamental law governing this notion of order is that if a b. We propose to investigate the common divisors of a and b.If a is divisible by b, then the common divisors of a and b consist simply of all divisors of b, and there is no more to be said. If a is not divisible by b, we can express a as a multiple of b together with a remainder less than b, that is a = qb + c, where c < b. (2) This is the process of ‘division with a remainder’, and expresses the fact that a, not being a multiple of b, must occur somewhere between two consecutive multiples of b. If a comes between qb and (q + 1)b, then a = qb + c, where 0 < c < b. It follows from the equation (2) that any common divisor of b and c is also a divisor of a.Moreo ver, any common divisor of a and b is also a divisor of c, since c = a ? qb. It follows that the common divisors of a and b, whatever they may be, are the same as the common divisors of b and c. The problem of ? nding the common divisors of a and b is reduced to the same problem for the numbers b and c, which are respectively less than a and b. The essence of the algorithm lies in the repetition of this argument. If b is divisible by c, the common divisors of b and c consist of all divisors of c. If not, we express b as b = r c + d, where d < c. (3)Again, the common divisors of b and c are the same as those of c and d. The process goes on until it terminates, and this can only happen when exact divisibility occurs, that is, when we come to a number in the sequence a, b, c, . . . , which is a divisor of the preceding number. It is plain that the process must terminate, for the decreasing sequence a, b, c, . . . of natural numbers cannot go on for ever. Factorization and the Primes 17 Let us suppose, for the sake of de? niteness, that the process terminates when we reach the number h, which is a divisor of the preceding number g.Then the last two equations of the series (2), (3), . . . are f = vg + h, g = wh. (4) (5) The common divisors of a and b are the same as those of b and c, or of c and d, and so on until we reach g and h. Since h divides g, the common divisors of g and h consist simply of all divisors of h. The number h can be identi? ed as being the last remainder in Euclid’s algorithm before exact divisibility occurs, i. e. the last non-zero remainder. We have therefore proved that the common divisors of two given natural numbers a and b consist of all divisors of a certain number h (the H. C. F. f a and b), and this number is the last non-zero remainder when Euclid’s algorithm is applied to a and b. As a numerical illustration, take the numbers 3132 and 7200 which were used in  §5. The algorithm runs as follows: 7200 = 2 ? 3132 + 936, 3 132 = 3 ? 936 + 324, 936 = 2 ? 324 + 288, 324 = 1 ? 288 + 36, 288 = 8 ? 36; and the H. C. F. is 36, the last remainder. It is often possible to shorten the working a little by using a negative remainder whenever this is numerically less than the corresponding positive remainder. In the above example, the last three steps could be replaced by 936 = 3 ? 324 ? 6, 324 = 9 ? 36. The reason why it is permissible to use negative remainders is that the argument that was applied to the equation (2) would be equally valid if that equation were a = qb ? c instead of a = qb + c. Two numbers are said to be relatively prime? if they have no common divisor except 1, or in other words if their H. C. F. is 1. This will be the case if and only if the last remainder, when Euclid’s algorithm is applied to the two numbers, is 1. ? This is, of course, the same de? nition as in  §5, but is repeated here because the present treatment is independent of that given previously. 8 7. Another proof of t he fundamental theorem The Higher Arithmetic We shall now use Euclid’s algorithm to give another proof of the fundamental theorem of arithmetic, independent of that given in  §4. We begin with a very simple remark, which may be thought to be too obvious to be worth making. Let a, b, n be any natural numbers. The highest common factor of na and nb is n times the highest common factor of a and b. However obvious this may seem, the reader will ? nd that it is not easy to give a proof of it without using either Euclid’s algorithm or the fundamental theorem of arithmetic.In fact the result follows at once from Euclid’s algorithm. We can suppose a > b. If we divide na by nb, the quotient is the same as before (namely q) and the remainder is nc instead of c. The equation (2) is replaced by na = q. nb + nc. The same applies to the later equations; they are all simply multiplied throughout by n. Finally, the last remainder, giving the H. C. F. of na and nb, is nh, wher e h is the H. C. F. of a and b. We apply this simple fact to prove the following theorem, often called Euclid’s theorem, since it occurs as Prop. 30 of Book VII.If a prime divides the product of two numbers, it must divide one of the numbers (or possibly both of them). Suppose the prime p divides the product na of two numbers, and does not divide a. The only factors of p are 1 and p, and therefore the only common factor of p and a is 1. Hence, by the theorem just proved, the H. C. F. of np and na is n. Now p divides np obviously, and divides na by hypothesis. Hence p is a common factor of np and na, and so is a factor of n, since we know that every common factor of two numbers is necessarily a factor of their H. C. F.We have therefore proved that if p divides na, and does not divide a, it must divide n; and this is Euclid’s theorem. The uniqueness of factorization into primes now follows. For suppose a number n has two factorizations, say n = pqr . . . = p q r . . . , where all the numbers p, q, r, . . . , p , q , r , . . . are primes. Since p divides the product p (q r . . . ) it must divide either p or q r . . . . If p divides p then p = p since both numbers are primes. If p divides q r . . . we repeat the argument, and ultimately reach the conclusion that p must equal one of the primes p , q , r , . . . We can cancel the common prime p from the two representations, and start again with one of those left, say q. Eventually it follows that all the primes on the left are the same as those on the right, and the two representations are the same. Factorization and the Primes 19 This is the alternative proof of the uniqueness of factorization into primes, which was referred to in  §4. It has the merit of resting on a general theory (that of Euclid’s algorithm) rather than on a special device such as that used in  §4. On the other hand, it is longer and less direct. 8. A property of the H. C.F From Euclid’s algorithm one can deduce a remarkable property of the H. C. F. , which is not at all apparent from the original construction for the H. C. F. by factorization into primes ( §5). The property is that the highest common factor h of two natural numbers a and b is representable as the difference between a multiple of a and a multiple of b, that is h = ax ? by where x and y are natural numbers. Since a and b are both multiples of h, any number of the form ax ? by is necessarily a multiple of h; and what the result asserts is that there are some values of x and y for which ax ? y is actually equal to h. Before giving the proof, it is convenient to note some properties of numbers representable as ax ? by. In the ? rst place, a number so representable can also be represented as by ? ax , where x and y are natural numbers. For the two expressions will be equal if a(x + x ) = b(y + y ); and this can be ensured by taking any number m and de? ning x and y by x + x = mb, y + y = ma. These numbers x and y will be natura l numbers provided m is suf? ciently large, so that mb > x and ma > y. If x and y are de? ned in this way, then ax ? by = by ? x . We say that a number is linearly dependent on a and b if it is representable as ax ? by. The result just proved shows that linear dependence on a and b is not affected by interchanging a and b. There are two further simple facts about linear dependence. If a number is linearly dependent on a and b, then so is any multiple of that number, for k(ax ? by) = a. kx ? b. ky. Also the sum of two numbers that are each linearly dependent on a and b is itself linearly dependent on a and b, since (ax1 ? by1 ) + (ax2 ? by2 ) = a(x1 + x2 ) ? b(y1 + y2 ). 20 The Higher ArithmeticThe same applies to the difference of two numbers: to see this, write the second number as by2 ? ax2 , in accordance with the earlier remark, before subtracting it. Then we get (ax1 ? by1 ) ? (by2 ? ax2 ) = a(x1 + x2 ) ? b(y1 + y2 ). So the property of linear dependence on a and b is preserved by addition and subtraction, and by multiplication by any number. We now examine the steps in Euclid’s algorithm, in the light of this concept. The numbers a and b themselves are certainly linearly dependent on a and b, since a = a(b + 1) ? b(a), b = a(b) ? b(a ? 1). The ? rst equation of the algorithm was a = qb + c.Since b is linearly dependent on a and b, so is qb, and since a is also linearly dependent on a and b, so is a ? qb, that is c. Now the next equation of the algorithm allows us to deduce in the same way that d is linearly dependent on a and b, and so on until we come to the last remainder, which is h. This proves that h is linearly dependent on a and b, as asserted. As an illustration, take the same example as was used in  §6, namely a = 7200 and b = 3132. We work through the equations one at a time, using them to express each remainder in terms of a and b. The ? rst equation was 7200 = 2 ? 3132 + 936, which tells s that 936 = a ? 2b. The second equation was 3 132 = 3 ? 936 + 324, which gives 324 = b ? 3(a ? 2b) = 7b ? 3a. The third equation was 936 = 2 ? 324 + 288, which gives 288 = (a ? 2b) ? 2(7b ? 3a) = 7a ? 16b. The fourth equation was 324 = 1 ? 288 + 36, Factorization and the Primes which gives 36 = (7b ? 3a) ? (7a ? 16b) = 23b ? 10a. 21 This expresses the highest common factor, 36, as the difference of two multiples of the numbers a and b. If one prefers an expression in which the multiple of a comes ? rst, this can be obtained by arguing that 23b ? 10a = (M ? 10)a ? (N ? 23)b, provided that Ma = N b.Since a and b have the common factor 36, this factor can be removed from both of them, and the condition on M and N becomes 200M = 87N . The simplest choice for M and N is M = 87, N = 200, which on substitution gives 36 = 77a ? 177b. Returning to the general theory, we can express the result in another form. Suppose a, b, n are given natural numbers, and it is desired to ? nd natural numbers x and y such that ax ? by = n. (6) Such an e quation is called an indeterminate equation since it does not determine x and y completely, or a Diophantine equation after Diophantus of Alexandria (third century A . D . , who wrote a famous treatise on arithmetic. The equation (6) cannot be soluble unless n is a multiple of the highest common factor h of a and b; for this highest common factor divides ax ? by, whatever values x and y may have. Now suppose that n is a multiple of h, say, n = mh. Then we can solve the equation; for all we have to do is ? rst solve the equation ax1 ? by1 = h, as we have seen how to do above, and then multiply throughout by m, getting the solution x = mx1 , y = my1 for the equation (6). Hence the linear indeterminate equation (6) is soluble in natural numbers x, y if and only if n is a multiple of h.In particular, if a and b are relatively prime, so that h = 1, the equation is soluble whatever value n may have. As regards the linear indeterminate equation ax + by = n, we have found the condition for it to be soluble, not in natural numbers, but in integers of opposite signs: one positive and one negative. The question of when this equation is soluble in natural numbers is a more dif? cult one, and one that cannot well be completely answered in any simple way. Certainly 22 The Higher Arithmetic n must be a multiple of h, but also n must not be too small in relation to a and b.It can be proved quite easily that the equation is soluble in natural numbers if n is a multiple of h and n > ab. 9. Factorizing a number The obvious way of factorizing a number is to test whether it is divisible by 2 or by 3 or by 5, and so on, using the series of primes. If a number N v is not divisible by any prime up to N , it must be itself a prime; for any composite number has at least two prime factors, and they cannot both be v greater than N . The process is a very laborious one if the number is at all large, and for this reason factor tables have been computed.The most extensive one which is gener ally accessible is that of D. N. Lehmer (Carnegie Institute, Washington, Pub. No. 105. 1909; reprinted by Hafner Press, New York, 1956), which gives the least prime factor of each number up to 10,000,000. When the least prime factor of a particular number is known, this can be divided out, and repetition of the process gives eventually the complete factorization of the number into primes. Several mathematicians, among them Fermat and Gauss, have invented methods for reducing the amount of trial that is necessary to factorize a large number.Most of these involve more knowledge of number-theory than we can postulate at this stage; but there is one method of Fermat which is in principle extremely simple and can be explained in a few words. Let N be the given number, and let m be the least number for which m 2 > N . Form the numbers m 2 ? N , (m + 1)2 ? N , (m + 2)2 ? N , . . . . (7) When one of these is reached which is a perfect square, we get x 2 ? N = y 2 , and consequently N = x 2 ? y 2 = (x ? y)(x + y). The calculation of the numbers (7) is facilitated by noting that their successive differences increase at a constant rate. The identi? ation of one of them as a perfect square is most easily made by using Barlow’s Table of Squares. The method is particularly successful if the number N has a factorization in which the two factors are of about the same magnitude, since then y is small. If N is itself a prime, the process goes on until we reach the solution provided by x + y = N , x ? y = 1. As an illustration, take N = 9271. This comes between 962 and 972 , so that m = 97. The ? rst number in the series (7) is 972 ? 9271 = 138. The Factorization and the Primes 23 subsequent ones are obtained by adding successively 2m + 1, then 2m + 3, and so on, that is, 195, 197, and so on.This gives the series 138, 333, 530, 729, 930, . . . . The fourth of these is a perfect square, namely 272 , and we get 9271 = 1002 ? 272 = 127 ? 73. An interesting algorithm for fact orization has been discovered recently by Captain N. A. Draim, U . S . N. In this, the result of each trial division is used to modify the number in preparation for the next division. There are several forms of the algorithm, but perhaps the simplest is that in which the successive divisors are the odd numbers 3, 5, 7, 9, . . . , whether prime or not. To explain the rules, we work a numerical example, say N = 4511. The ? st step is to divide by 3, the quotient being 1503 and the remainder 2: 4511 = 3 ? 1503 + 2. The next step is to subtract twice the quotient from the given number, and then add the remainder: 4511 ? 2 ? 1503 = 1505, 1505 + 2 = 1507. The last number is the one which is to be divided by the next odd number, 5: 1507 = 5 ? 301 + 2. The next step is to subtract twice the quotient from the ? rst derived number on the previous line (1505 in this case), and then add the remainder from the last line: 1505 ? 2 ? 301 = 903, 903 + 2 = 905. This is the number which is to be divi ded by the next odd number, 7. Now we an continue in exactly the same way, and no further explanation will be needed: 905 = 7 ? 129 + 2, 903 ? 2 ? 129 = 645, 645 ? 2 ? 71 = 503, 503 ? 2 ? 46 = 411, 645 + 2 = 647, 503 + 8 = 511, 411 + 5 = 416, 647 = 9 ? 71 + 8, 511 = 11 ? 46 + 5, 416 = 13 ? 32 + 0. 24 The Higher Arithmetic We have reached a zero remainder, and the algorithm tells us that 13 is a factor of the given number 4511. The complementary factor is found by carrying out the ? rst half of the next step: 411 ? 2 ? 32 = 347. In fact 4511 = 13? 347, and as 347 is a prime the factorization is complete. To justify the algorithm generally is a matter of elementary algebra.Let N1 be the given number; the ? rst step was to express N1 as N1 = 3q1 + r1 . The next step was to form the numbers M2 = N1 ? 2q1 , The number N2 was divided by 5: N2 = 5q2 + r2 , and the next step was to form the numbers M3 = M2 ? 2q2 , N 3 = M3 + r 2 , N 2 = M2 + r 1 . and so the process was continued. It can be deduced from these equations that N2 = 2N1 ? 5q1 , N3 = 3N1 ? 7q1 ? 7q2 , N4 = 4N1 ? 9q1 ? 9q2 ? 9q3 , and so on. Hence N2 is divisible by 5 if and only if 2N1 is divisible by 5, or N1 divisible by 5. Again, N3 is divisible by 7 if and only if 3N1 is divisible by 7, or N1 divisible by 7, and so on.When we reach as divisor the least prime factor of N1 , exact divisibility occurs and there is a zero remainder. The general equation analogous to those given above is Nn = n N1 ? (2n + 1)(q1 + q2 +  ·  ·  · + qn? 1 ). The general equation for Mn is found to be Mn = N1 ? 2(q1 + q2 +  ·  ·  · + qn? 1 ). (9) If 2n + 1 is a factor of the given number N1 , then Nn is exactly divisible by 2n + 1, and Nn = (2n + 1)qn , whence n N1 = (2n + 1)(q1 + q2 +  ·  ·  · + qn ), (8) Factorization and the Primes by (8). Under these circumstances, we have, by (9), Mn+1 = N1 ? 2(q1 + q2 +  ·  ·  · + qn ) = N1 ? 2 n 2n + 1 N1 = N1 . n + 1 25 Thus the complementary factor to the factor 2n + 1 is Mn+1 , as stated in the example. In the numerical example worked out above, the numbers N1 , N2 , . . . decrease steadily. This is always the case at the beginning of the algorithm, but may not be so later. However, it appears that the later numbers are always considerably less than the original number. 10. The series of primes Although the notion of a prime is a very natural and obvious one, questions concerning the primes are often very dif? cult, and many such questions are quite unanswerable in the present state of mathematical knowledge.We conclude this chapter by mentioning brie? y some results and conjectures about the primes. In  §3 we gave Euclid’s proof that there are in? nitely many primes. The same argument will also serve to prove that there are in? nitely many primes of certain speci? ed forms. Since every prime after 2 is odd, each of them falls into one of the two progressions (a) 1, 5, 9, 13, 17, 21, 25, . . . , (b) 3, 7, 11, 15, 19, 23, 27, . . . ; the progression (a) consisting of all numbers of the form 4x + 1, and the progression (b) of all numbers of the form 4x ? 1 (or 4x + 3, which comes to the same thing).We ? rst prove that there are in? nitely many primes in the progression (b). Let the primes in (b) be enumerated as q1 , q2 , . . . , beginning with q1 = 3. Consider the number N de? ned by N = 4(q1 q2 . . . qn ) ? 1. This is itself a number of the form 4x ? 1. Not every prime factor of N can be of the form 4x + 1, because any product of numbers which are all of the form 4x + 1 is itself of that form, e. g. (4x + 1)(4y + 1) = 4(4x y + x + y) + 1. Hence the number N has some prime factor of the form 4x ? 1. This cannot be any of the primes q1 , q2 , . . . , qn , since N leaves the remainder ? when 26 The Higher Arithmetic divided by any of them. Thus there exists a prime in the series (b) which is different from any of q1 , q2 , . . . , qn ; and this proves the proposition. The same argument cannot be used to prove t hat there are in? nitely many primes in the series (a), because if we construct a number of the form 4x +1 it does not follow that this number will necessarily have a prime factor of that form. However, another argument can be used. Let the primes in the series (a) be enumerated as r1 , r2 , . . . , and consider the number M de? ned by M = (r1 r2 . . rn )2 + 1. We shall see later (III. 3) that any number of the form a 2 + 1 has a prime factor of the form 4x + 1, and is indeed entirely composed of such primes, together possibly with the prime 2. Since M is obviously not divisible by any of the primes r1 , r2 , . . . , rn , it follows as before that there are in? nitely many primes in the progression (a). A similar situation arises with the two progressions 6x + 1 and 6x ? 1. These progressions exhaust all numbers that are not divisible by 2 or 3, and therefore every prime after 3 falls in one of these two progressions.One can prove by methods similar to those used above that there ar e in? nitely many primes in each of them. But such methods cannot cope with the general arithmetical progression. Such a progression consists of all numbers ax +b, where a and b are ? xed and x = 0, 1, 2, . . . , that is, the numbers b, b + a, b + 2a, . . . . If a and b have a common factor, every number of the progression has this factor, and so is not a prime (apart from possibly the ? rst number b). We must therefore suppose that a and b are relatively prime. It then seems plausible that the progression will contain in? itely many primes, i. e. that if a and b are relatively prime, there are in? nitely many primes of the form ax + b. Legendre seems to have been the ? rst to realize the importance of this proposition. At one time he thought he had a proof, but this turned out to be fallacious. The ? rst proof was given by Dirichlet in an important memoir which appeared in 1837. This proof used analytical methods (functions of a continuous variable, limits, and in? nite series), an d was the ? rst really important application of such methods to the theory of numbers.It opened up completely new lines of development; the ideas underlying Dirichlet’s argument are of a very general character and have been fundamental for much subsequent work applying analytical methods to the theory of numbers. Factorization and the Primes 27 Not much is known about other forms which represent in? nitely many primes. It is conjectured, for instance, that there are in? nitely many primes of the form x 2 + 1, the ? rst few being 2, 5, 17, 37, 101, 197, 257, . . . . But not the slightest progress has been made towards proving this, and the question seems hopelessly dif? cult.Dirichlet did succeed, however, in proving that any quadratic form in two variables, that is, any form ax 2 + bx y + cy 2 , in which a, b, c are relatively prime, represents in? nitely many primes. A question which has been deeply investigated in modern times is that of the frequency of occurrence of the p rimes, in other words the question of how many primes there are among the numbers 1, 2, . . . , X when X is large. This number, which depends of course on X , is usually denoted by ? (X ). The ? rst conjecture about the magnitude of ? (X ) as a function of X seems to have been made independently by Legendre and Gauss about X 1800.It was that ? (X ) is approximately log X . Here log X denotes the natural (so-called Napierian) logarithm of X , that is, the logarithm of X to the base e. The conjecture seems to have been based on numerical evidence. For example, when X is 1,000,000 it is found that ? (1,000,000) = 78,498, whereas the value of X/ log X (to the nearest integer) is 72,382, the ratio being 1. 084 . . . . Numerical evidence of this kind may, of course, be quite misleading. But here the result suggested is true, in the sense that the ratio of ? (X ) to X/ log X tends to the limit 1 as X tends to in? ity. This is the famous Prime Number Theorem, ? rst proved by Hadamard and de la Vall? e e Poussin independently in 1896, by the use of new and powerful analytical methods. It is impossible to give an account here of the many other results which have been proved concerning the distribution of the primes. Those proved in the nineteenth century were mostly in the nature of imperfect approaches towards the Prime Number Theorem; those of the twentieth century included various re? nements of that theorem. There is one recent event to which, however, reference should be made.We have already said that the proof of Dirichlet’s Theorem on primes in arithmetical progressions and the proof of the Prime Number Theorem were analytical, and made use of methods which cannot be said to belong properly to the theory of numbers. The propositions themselves relate entirely to the natural numbers, and it seems reasonable that they should be provable without the intervention of such foreign ideas. The search for ‘elementary’ proofs of these two theorems was u nsuccessful until fairly recently. In 1948 A. Selberg found the ? rst elementary proof of Dirichlet’s Theorem, and with 28 The Higher Arithmetic he help of P. Erd? s he found the ? rst elementary proof of the Prime Numo ber Theorem. An ‘elementary’ proof, in this connection, means a proof which operates only with natural numbers. Such a proof is not necessarily simple, and indeed both the proofs in question are distinctly dif? cult. Finally, we may mention the famous problem concerning primes which was propounded by Goldbach in a letter to Euler in 1742. Goldbach suggested (in a slightly different wording) that every even number from 6 onwards is representable as the sum of two primes other than 2, e. g. 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 12 = 5 + 7, . . . Any problem like this which relates to additive properties of primes is necessarily dif? cult, since the de? nition of a prime and the natural properties of primes are all expressed in terms of multiplic ation. An important contribution to the subject was made by Hardy and Littlewood in 1923, but it was not until 1930 that anything was rigorously proved that could be considered as even a remote approach towards a solution of Goldbach’s problem. In that year the Russian mathematician Schnirelmann proved that there is some number N such that every number from some point onwards is representable as the sum of at most N primes.A much nearer approach was made by Vinogradov in 1937. He proved, by analytical methods of extreme subtlety, that every odd number from some point onwards is representable as the sum of three primes. This was the starting point of much new work on the additive theory of primes, in the course of which many problems have been solved which would have been quite beyond the scope of any pre-Vinogradov methods. A recent result in connection with Goldbach’s problem is that every suf? ciently large even number is representable as the sum of two numbers, one of which is a prime and the other of which has at most two prime factors.Notes Where material is changing more rapidly than print cycles permit, we have chosen to place some of the material on the book’s website: www. cambridge. org/davenport. Symbols such as  ¦I:0 are used to indicate where there is such additional material.  §1. The main dif? culty in giving any account of the laws of arithmetic, such as that given here, lies in deciding which of the various concepts should come ? rst. There are several possible arrangements, and it seems to be a matter of taste which one prefers. It is no part of our purpose to analyse further the concepts and laws of ? rithmetic. We take the commonsense (or na? ve) view that we all ‘know’ Factorization and the Primes 29 the natural numbers, and are satis? ed of the validity of the laws of arithmetic and of the principle of induction. The reader who is interested in the foundations of mathematics may consult Bertrand Russe ll, Introduction to Mathematical Philosophy (Allen and Unwin, London), or M. Black, The Nature of Mathematics (Harcourt, Brace, New York). Russell de? nes the natural numbers by selecting them from numbers of a more general kind. These more general numbers are the (? ite or in? nite) cardinal numbers, which are de? ned by means of the more general notions of ‘class’ and ‘one-to-one correspondence’. The selection is made by de? ning the natural numbers as those which possess all the inductive properties. (Russell, loc. cit. , p. 27). But whether it is reasonable to base the theory of the natural numbers on such a vague and unsatisfactory concept as that of a class is a matter of opinion. ‘Dolus latet in universalibus’ as Dr Johnson remarked.  §2. The objection to using the principle of induction as a de? ition of the natural numbers is that it involves references to ‘any proposition about a natural number n’. It seems plain the th at ‘propositions’ envisaged here must be statements which are signi? cant when made about natural numbers. It is not clear how this signi? cance can be tested or appreciated except by one who already knows the natural numbers.  §4. I am not aware of having seen this proof of the uniqueness of prime factorization elsewhere, but it is unlikely that it is new. For other direct proofs, see Mathews, p. 2, or Hardy and Wright, p. 21.?  §5. It has been shown by (intelligent! computer searches that there is no odd perfect number less than 10300 . If an odd perfect number exists, it has at least eight distinct prime factors, of which the largest exceeds 108 . For references and other information on perfect or ‘nearly perfect’ numbers, see Guy, sections A. 3, B. 1 and B. 2.  ¦I:1  §6. A critical reader may notice that in two places in this section I have used principles that were not explicitly stated in  §Ã‚ §1 and 2. In each place, a proof by induction co uld have been given, but to have done so would have distracted the reader’s attention from the main issues.The question of the length of Euclid’s algorithm is discussed in Uspensky and Heaslet, ch. 3, and D. E. Knuth’s The Art of Computer Programming vol. II: Seminumerical Algorithms (Addison Wesley, Reading, Mass. , 3rd. ed. , 1998) section 4. 5. 3.  §9. For an account of early methods of factoring, see Dickson’s History Vol. I, ch. 14. For a discussion of the subject as it appeared in ? Particulars of books referred to by their authors’ names will be found in the Bibliography. 30 The Higher Arithmetic the 1970s see the article by Richard K. Guy, ‘How to factor a number’, Congressus Numerantium XVI Proc. th Manitoba Conf. Numer. Math. , Winnipeg, 1975, 49–89, and at the turn of the millennium see Richard P. Brent, ‘Recent progress and prospects for integer factorisation algorithms’, Springer Lecture Notes in Comp uter Science 1858 Proc. Computing and Combinatorics, 2000, 3–22. The subject is discussed further in Chapter VIII. It is doubtful whether D. N. Lehmer’s tables will ever be extended, since with them and a pocket calculator one can easily check whether a 12-digit number is a prime. Primality testing is discussed in VIII. 2 and VIII. 9. For Draim’s algorithm, see Mathematics Magazine, 25 (1952) 191–4. 10. An excellent account of the distribution of primes is given by A. E. Ingham, The Distribution of Prime Numbers (Cambridge Tracts, no. 30, 1932; reprinted by Hafner Press, New York, 1971). For a more recent and extensive account see H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). H. Iwaniec (Inventiones Math. 47 (1978) 171–88) has shown that for in? nitely many n the number n 2 + 1 is either prime or the product of at most two primes, and indeed the same is true for any irreducible an 2 + bn + c with c odd. Dirichlet’s p roof of his theorem (with a modi? ation due to Mertens) is given as an appendix to Dickson’s Modern Elementary Theory of Numbers. An elementary proof of the Prime Number Theorem is given in ch. 22 of Hardy and Wright. An elementary proof of the asymptotic formula for the number of primes in an arithmetic progression is given in Gelfond and Linnik, ch. 3. For a survey of early work on Goldbach’s problem, see James, Bull. American Math. Soc. , 55 (1949) 246–60. It has been veri? ed that every even number from 6 to 4 ? 1014 is the sum of two primes, see Richstein, Math. Comp. , 70 (2001) 1745–9. For a proof of Chen’s theorem that every suf? iently large even integer can be represented as p + P2 , where p is a prime, and P2 is either a prime or the product of two primes, see ch. 11 of Sieve Methods by H. Halberstam and H. E. Richert (Academic Press, London, 1974). For a proof of Vinogradov’s result, see T. Estermann, Introduction to Modern Prime Number Theory (Cambridge Tracts, no. 41, 1952) or H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). ‘Suf? ciently large’ in Vinogradov’s result has now been quanti? ed as ‘greater than 2 ? 101346 ’, see M. -C. Liu and T. Wang, Acta Arith. , 105 (2002) 133–175.Conversely, we know that it is true up to 1. 13256 ? 1022 (Ramar? and Saouter in J. Number Theory 98 (2003) 10–33). e II CONGRUENCES 1. The congruence notation It often happens that for the purposes of a particular calculation, two numbers which differ by a multiple of some ? xed number are equivalent, in the sense that they produce the same result. For example, the value of (? 1)n depends only on whether n is odd or even, so that two values of n which differ by a multiple of 2 give the same result. Or again, if we are concerned only with the last digit of a number, then for that purpose two umbers which differ by a multiple of 10 are effectively the same. The congruence notation, introduced by Gauss, serves to express in a convenient form the fact that two integers a and b differ by a multiple of a ? xed natural number m. We say that a is congruent to b with respect to the modulus m, or, in symbols, a ? b (mod m). The meaning of this, then, is simply that a ? b is divisible by m. The notation facilitates calculations in which numbers differing by a multiple of m are effectively the same, by stressing the analogy between congruence and equality.Congruence, in fact, means ‘equality except for the addition of some multiple of m’. A few examples of valid congruences are: 63 ? 0 (mod 3), 7 ? ?1 (mod 8), 52 ? ?1 (mod 13). A congruence to the modulus 1 is always valid, whatever the two numbers may be, since every number is a multiple of 1. Two numbers are congruent with respect to the modulus 2 if they are of the same parity, that is, both even or both odd. 31 32 The Higher Arithmetic Two congruences can be added, subtracted, or m ultiplied together, in just the same way as two equations, provided all the congruences have the same modulus.If a ? ? (mod m) and b ? ? (mod m) then a + b ? ? + ? (mod m), a ? b ? ? ? ? (mod m), ab ? (mod m). The ? rst two of these statements are immediate; for example (a + b) ? (? + ? ) is a multiple of m because a ? ? and b ? ? are both multiples of m. The third is not quite so immediate and is best proved in two steps. First ab ? ?b because ab ? ?b = (a ? ?)b, and a ? ? is a multiple of m. Next, ? b ? , for a similar reason. Hence ab ? (mod m). A congruence can always be multiplied throughout by any integer: if a ? ? (mod m) then ka ? k? (mod m).Indeed this is a special case of the third result above, where b and ? are both k. But it is not always legitimate to cancel a factor from a congruence. For example 42 ? 12 (mod 10), but it is not permissible to cancel the factor 6 from the numbers 42 and 12, since this would give the false result 7 ? 2 (mod 10). The reason is obvious : the ? rst congruence states that 42 ? 12 is a multiple of 10, but this does not imply that 1 (42 ? 12) is a multiple of 10. The cancellation of 6 a factor from a congruence is legitimate if the factor is relatively prime to the modulus.For let the given congruence be ax ? ay (mod m), where a is the factor to be cancelled, and we suppose that a is relatively prime to m. The congruence states that a(x ? y) is divisible by m, and it follows from the last proposition in I. 5 that x ? y is divisible by m. An illustration of the use of congruences is provided by the well-known rules for the divisibility of a number by 3 or 9 or 11. The usual representation of a number n by digits in the scale of 10 is really a representation of n in the form n = a + 10b + 100c +  ·  ·  · , where a, b, c, . . . re the digits of the number, read from right to left, so that a is the number of units, b the number of tens, and so on. Since 10 ? 1 (mod 9), we have also 102 ? 1 (mod 9), 103 ? 1 (mod 9), and so on. Hence it follows from the above representation of n that n ? a + b + c +  ·  ·  · (mod 9). Congruences 33 In other words, any number n differs from the sum of its digits by a multiple of 9, and in particular n is divisible by 9 if and only if the sum of its digits is divisible by 9. The same applies with 3 in place of 9 throughout. The rule for 11 is based on the fact that 10 ? ?1 (mod 11), so that 102 ? +1 (mod 11), 103 ? 1 (mod 11), and so on. Hence n ? a ? b + c ?  ·  ·  · (mod 11). It follows that n is divisible by 11 if and only if a ? b+c?  ·  ·  · is divisible by 11. For example, to test the divisibility of 9581 by 11 we form 1? 8+5? 9, or ? 11. Since this is divisible by 11, so is 9581. 2. Linear congruences It is obvious that every integer is congruent (mod m) to exactly one of the numbers 0, 1, 2, . . . , m ? 1. (1) r < m, For we can express the integer in the form qm + r , where 0 and then it is congruent to r (mod m). Obviously there are othe r sets of numbers, besides the set (1), which have the same property, e. . any integer is congruent (mod 5) to exactly one of the numbers 0, 1, ? 1, 2, ? 2. Any such set of numbers is said to constitute a complete set of residues to the modulus m. Another way of expressing the de? nition is to say that a complete set of residues (mod m) is any set of m numbers, no two of which are congruent to one another. A linear congruence, by analogy with a linear equation in elementary algebra, means a congruence of the form ax ? b (mod m). (2) It is an important fact that any such congruence is soluble for x, provided that a is relatively prime to m.The simplest way of proving this is to observe that if x runs through the numbers of a complete set of residues, then the corresponding values of ax also constitute a complete set of residues. For there are m of these numbers, and no two of them are congruent, since ax 1 ? ax2 (mod m) would involve x1 ? x2 (mod m), by the cancellation of the factor a (permissible since a is relatively prime to m). Since the numbers ax form a complete set of residues, there will be exactly one of them congruent to the given number b. As an example, consider the congruence 3x ? 5 (mod 11). 34 The Higher ArithmeticIf we give x the values 0, 1, 2, . . . , 10 (a complete set of residues to the modulus 11), 3x takes the values 0, 3, 6, . . . , 30. These form another complete set of residues (mod 11), and in fact they are congruent respectively to 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8. The value 5 occurs when x = 9, and so x = 9 is a solution of the congruence. Naturally any number congruent to 9 (mod 11) will also satisfy the congruence; but nevertheless we say that the congruence has one solution, meaning that there is one solution in any complete set of residues. In other words, all solutions are mutually congruent.The same applies to the general congruence (2); such a congruence (provided a is relatively prime to m) is precisely equivalent to the con gruence x ? x0 (mod m), where x0 is one particular solution. There is another way of looking at the linear congruence (2). It is equivalent to the equation ax = b + my, or ax ? my = b. We proved in I. 8 that such a linear Diophantine equation is soluble for x and y if a and m are relatively prime, and that fact provides another proof of the solubility of the linear congruence. But the proof given above is simpler, and illustrates the advantages gained by using the congruence notation.The fact that the congruence (2) has a unique solution, in the sense explained above, suggests that one may use this solution as an interpretation b for the fraction a to the modulus m. When we do this, we obtain an arithmetic (mod m) in which addition, subtraction and multiplication are always possible, and division is also possible provided that the divisor is relatively prime to m. In this arithmetic there are only a ? nite number of essentially distinct numbers, namely m of them, since two numbers w hich are mutually congruent (mod m) are treated as the same.If we take the modulus m to be 11, as an illustration, a few examples of ‘arithmetic mod 11’ are: 5 ? 9 ? ?2. 3 Any relation connecting integers or fractions in the ordinary sense remains true when interpreted in this arithmetic. For example, the relation 5 + 7 ? 1, 5 ? 6 ? 8, 1 2 7 + = 2 3 6 becomes (mod 11) 6 + 8 ? 3, because the solution of 2x ? 1 is x ? 6, that of 3x ? 2 is x ? 8, and that of 6x ? 7 is x ? 3. Naturally the interpretation given to a fraction depends on the modulus, for instance 2 ? 8 (mod 11), but 2 ? 3 (mod 7). The 3 3 Congruences 35 nly limitation on such calculations is that just mentioned, namely that the denominator of any fraction must be relatively prime to the modulus. If the modulus is a prime (as in the above examples with 11), the limitation takes the very simple form that the denominator must not be congruent to 0 (mod m), and this is exactly analogous to the limitation in ordina ry arithmetic that the denominator must not be equal to 0. We shall return to this point later ( §7). 3. Fermat’s theorem The fact that there are only a ? nite number of essentially different numbers in arithmetic to a modulus m means that there are algebraic relations which are satis? d by every number in that arithmetic. There is nothing analogous to these relations in ordinary arithmetic. Suppose we take any number x and consider its powers x, x 2 , x 3 , . . . . Since there are only a ? nite number of possibilities for these to the modulus m, we must eventually come to one which we have met before, say x h ? x k (mod m), where k < h. If x is relatively prime to m, the factor x k can be cancelled, and it follows that x l ? 1 (mod m), where l ? h ? k. Hence every number x which is relatively prime to m satis? es some congruence of this form. The least exponent l for which x l ? (mod m) will be called the order of x to the modulus m. If x is 1, its order is obviously 1. To illustrate the de? nition, let us calculate the orders of a few numbers to the modulus 11. The powers of 2, taken to the modulus 11, are 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, . . . . Each one is twice the preceding one, with 11 or a multiple of 11 subtracted where necessary to make the result less than 11. The ? rst power of 2 which is ? 1 is 210 , and so the order of 2 (mod 11) is 10. As another example, take the powers of 3: 3, 9, 5, 4, 1, 3, 9, . . . . The ? rst power of 3 which is ? 1 is 35 , so the order of 3 (mod 11) is 5.It will be found that the order of 4 is again 5, and so also is that of 5. It will be seen that the successive powers of x are periodic; when we have reached the ? rst number l for which x l ? 1, then x l+1 ? x and the previous cycle is repeated. It is plain that x n ? 1 (mod m) if and only if n is a multiple of the order of x. In the last example, 3n ? 1 (mod 11) if and only if n is a multiple of 5. This remains valid if n is 0 (since 30 = 1), and it remains valid also for negative exponents, provided 3? n , or 1/3n , is interpreted as a fraction (mod 11) in the way explained in  §2. 36 The Higher ArithmeticIn fact, the negative powers of 3 (mod 11) are obtained by prolonging the series backwards, and the table of powers of 3 to the modulus 11 is n =†¦ ?3 ? 2 ? 1 0 1 2 3 4 5 6 . . . 9 5 4 1 3 9 5 4 1 3 †¦ . 3n ? . . . Fermat discovered that if the modulus is a prime, say p, then every integer x not congruent to 0 satis? es x p? 1 ? 1 (mod p). (3) In view of what we have seen above, this is equivalent to saying that the order of any number is a divisor of p ? 1. The result (3) was mentioned by Fermat in a letter to Fr? nicle de Bessy of 18 October 1640, in which he e also stated that he had a proof.But as with most of Fermat’s discoveries, the proof was not published or preserved. The ? rst known proof seems to have been given by Leibniz (1646–1716). He proved that x p ? x (mod p), which is equivalent to (3), b y writing x as a sum 1 + 1 +  ·  ·  · + 1 of x units (assuming x positive), and then expanding (1 + 1 +  ·  ·  · + 1) p by the multinomial theorem. The terms 1 p + 1 p +  ·  ·  · + 1 p give x, and the coef? cients of all the other terms are easily proved to be divisible by p. Quite a different proof was given by Ivory in 1806. If x ? 0 (mod p), the integers x, 2x, 3x, . . . , ( p ? )x are congruent (in some order) to the numbers 1, 2, 3, . . . , p ? 1. In fact, each of these sets constitutes a complete set of residues except that 0 has been omitted from each. Since the two sets are congruent, their products are congruent, and so (x)(2x)(3x) . . . (( p ? 1)x) ? (1)(2)(3) . . . ( p ? 1)(mod p). Cancelling the factors 2, 3, . . . , p ? 1, as is permissible, we obtain (3). One merit of this proof is that it can be extended so as to apply to the more general case when the modulus is no longer a prime. The generalization of the result (3) to any modulus was ? rst given b y Euler in 1760.To formulate it, we must begin by considering how many numbers in the set 0, 1, 2, . . . , m ? 1 are relatively prime to m. Denote this number by ? (m). When m is a prime, all the numbers in the set except 0 are relatively prime to m, so that ? ( p) = p ? 1 for any prime p. Euler’s generalization of Fermat’s theorem is that for any modulus m, x ? (m) ? 1 (mod m), provided only that x is relatively prime to m. (4) Congruences 37 To prove this, it is only necessary to modify Ivory’s method by omitting from the numbers 0, 1, . . . , m ? 1 not only the number 0, but all numbers which are not relatively prime to m.There remain ? (m) numbers, say a 1 , a2 , . . . , a? , Then the numbers a1 x, a2 x, . . . , a? x are congruent, in some order, to the previous numbers, and on multiplying and cancelling a1 , a2 , . . . , a? (as is permissible) we obtain x ? ? 1 (mod m), which is (4). To illustrate this proof, take m = 20. The numbers less than 20 and relati vely prime to 20 are 1, 3, 7, 9, 11, 13, 17, 19, so that ? (20) = 8. If we multiply these by any number x which is relatively prime to 20, the new numbers are congruent to the original numbers in some other order.For example, if x is 3, the new numbers are congruent respectively to 3, 9, 1, 7, 13, 19, 11, 17 (mod 20); and the argument proves that 38 ? 1 (mod 20). In fact, 38 = 6561. where ? = ? (m). 4. Euler’s function ? (m) As we have just seen, this is the number of numbers up to m that are relatively prime to m. It is natural to ask what relation ? (m) bears to m. We saw that ? ( p) = p ? 1 for any prime p. It is also easy to evaluate ? ( p a ) for any prime power pa . The only numbers in the set 0, 1, 2, . . . , pa ? 1 which are not relatively prime to p are those that are divisible by p. These are the numbers pt, where t = 0, 1, . . , pa? 1 ? 1. The number of them is pa? 1 , and when we subtract this from the total number pa , we obtain ? ( pa ) = pa ? pa? 1 = pa? 1 ( p ? 1). (5) The determination of ? (m) for general values of m is effected by proving that this function is multiplicative. By this is meant that if a and b are any two relatively prime numbers, then ? (ab) = ? (a)? (b). (6) 38 The Higher Arithmetic To prove this, we begin by observing a general principle: if a and b are relatively prime, then two simultaneous congruences of the form x ? ? (mod a), x ? ? (mod b) (7) are precisely equivalent to one congruence to the modulus ab.For the ? rst congruence means that x = ? + at where t is an integer. This satis? es the second congruence if and only if ? + at ? ? (mod b), or at ? ? ? ? (mod b). This, being a linear congruence for t, is soluble. Hence the two congruences (7) are simultaneously soluble. If x and x are two solutions, we have x ? x (mod a) and x ? x (mod b), and therefore x ? x (mod ab). Thus there is exactly one solution to the modulus ab. This principle, which extends at once to several congruences, provided that the moduli ar e relatively prime in pairs, is sometimes called ‘the Chinese remainder theorem’.It assures us of the existence of numbers which leave prescribed remainders on division by the moduli in question. Let us represent the solution of the two congruences (7) by x ? [? , ? ] (mod ab), so that [? , ? ] is a certain number depending on ? and ? (and also on a and b of course) which is uniquely determined to the modulus ab. Different pairs of values of ? and ? give rise to different values for [? , ? ]. If we give ? the values 0, 1, . . . , a ? 1 (forming a complete set of residues to the modulus a) and similarly give ? the values 0, 1, . . . , b ? 1, the resulting values of [? , ? constitute a complete set of residues to the modulus ab. It is obvious that if ? has a factor in common with a, then x in (7) will also have that factor in common with a, in other words, [? , ? ] will have that factor in common with a. Thus [? , ? ] will only be relatively prime to ab if ? is relatively prime to a and ? is relatively prime to b, and conversely these conditions will ensure that [? , ? ] is relatively prime to ab. It follows that if we give ? the ? (a) possible values that are less than a and prime to a, and give ? the ? (b) values that are less than b and prime to b, there result ? (a)? (b) values of [? ? ], and these comprise all the numbers that are less than ab and relatively prime to ab. Hence ? (ab) = ? (a)? (b), as asserted in (6). To illustrate the situation arising in the above proof, we tabulate below the values of [? , ? ] when a = 5 and b = 8. The possible values for ? are 0, 1, 2, 3, 4, and the possible values for ? are 0, 1, 2, 3, 4, 5, 6, 7. Of these there are four values of ? which are relatively prime to a, corresponding to the fact that ? (5) = 4, and four values of ? that are relatively prime to b, Congruences 39 corresponding to the fact that ? (8) = 4, in accordance with the formula (5).These values are italicized, as also are the corresponding values of [? , ? ]. The latter constitute the sixteen numbers that are relatively prime to 40 and less than 40, thus verifying that ? (40) = ? (5)? (8) = 4 ? 4 = 16. ? ? 0 1 2 3 4 0 0 16 32 8 24 1 25 1 17 33 9 2 10 26 2 18 34 3 35 11 27 3 19 4 20 36 12 28 4 5 5 21 37 13 29 6 30 6 22 38 14 7 15 31 7 23 39 We now return to the original question, that of evaluating ? (m) for any number m. Suppose the factorization of m into prime powers is m = pa q b . . . . Then it follows from (5) and (6) that ? (m) = ( pa ? pa? 1 )(q b ? q b? 1 ) . . . or, more elegantly, ? (m) = m 1 ? For example, ? (40) = 40 1 ? and ? (60) = 60 1 ? 1 2 1 2 1 p 1? 1 q †¦. (8) 1? 1 3 1 5 = 16, 1 5 1? 1? = 16. The function ? (m) has a remarkable property, ? rst given by Gauss in his Disquisitiones. It is that the sum of the numbers ? (d), extended over all the divisors d of a number m, is equal to m itself. For example, if m = 12, the divisors are 1, 2, 3, 4, 6, 12, and we have ? (1) + ? (2) + ? (3) + ? (4) + ? (6) + ? (12) = 1 + 1 + 2 + 2 + 2 + 4 = 12. A general proof can be based either on (8), or directly on the de? nition of the function. 40 The Higher ArithmeticWe have already referred (I. 5) to a table of the values of ? (m) for m 10, 000. The same volume contains a table giving those numbers m for which ? (m) assumes a given value up to 2,500. This table shows that, up to that point at least, every value assumed by ? (m) is assumed at least twice. It seems reasonable to conjecture that this is true generally, in other words that for any number m there is another number m such that ? (m ) = ? (m). This has never been proved, and any attempt at a general proof seems to meet with formidable dif? culties. For some special types of numbers the result is easy, e. g. f m is odd, then ? (m) = ? (2m); or again if m is not divisible by 2 or 3 we have ? (3m) = ? (4m) = ? (6m). 5. Wilson’s theorem This theorem was ? rst publis