Eulers Totient Theorem

Eulers Totient TheoremSummary Euler Totient theorem is a generalized form of Fermats Little supposition. As such, it solely depends on Fermats Little Theorem as indicated in Eulers correction in 1763 and, later in 1883, the theorem was named after him by J. J. Sylvester. According to Sylvester, the theorem is basically about the alteration in similarity. The term Totient was derived from Quotient, hence, the function deals with division, just in a unique way. In this manner, The Eulers Totient function for any integer (n) disregard be demarcated, as the figure of positive integers is not greater than and co-prime to n.a(n) = 1 (mod n) found on Leonhard Eulers contributions toward the development of this theorem, the theory was named after him despite the fact that it was a generalization of Fermats Little Theory in which n is identified to be prime. found on this fact, some scholarly source refers to this theorem as the Fermats-Euler theorem of Eulers generalization.Introductio nI starting time developed an interest in Euler when I was completing a tender crossword the concealed message read Euler was the master of the crossword. When I first saw the inclusion of the name Euler on the list of prompt words, I had no option but to just go for it. Euler was a famous mathematician in the eighteenth century, who was acknowledged for his contribution in the math discipline, as he was trus bothrthy for proving numerous problems and conjectures. Taking the number theory as an example, Euler successively played a vital role in proving the two-squ are theorem as well as the Fermats small(a) theorem (Griffiths and Peter 81). His contribution also paved the way to proving the four-square theorem. Therefore, in this course project, I am going to focus on his theory, which is not known to umpteen it is a generalization of Fermats little theorem that is commonly known as Eulers theorem.TheoremEulers Totient theorem holds that if a and n are coprime positive integers , then since n is a Eulers Totient function.Eulers Totient solveEulers Totient Function (n) is the count of positive integers that are less that n and relatively prime to n. For instance, 10 is 4, since there are four integers, which are less than 10 and are relatively prime to 10 1, 3, 7, 9. Consequently, 11 is 10, since there 11 prime numbers which are less than 10 and are relatively prime to 10. The similar way, 6 is 2 as 1 and 5 are relatively prime to 6, but 2, 3, and 4 are not.The spare-time activity table represents the totients of numbers up to twenty.Nn21324254627684961041110124131214615816817161861918208Some of these examples seek to prove Eulers totient theorem.Let n = 10 and a = 3. In this case, 10 and 3 are co-prime i.e. relatively prime. Using the provided table, it is clear that 10 = 4. Then this relation can also be represented as follows34 = 81 1 (mode 10). Conversely, if n = 15 and a = 2, it is clear that 28 = 256 1 (mod 15).Fermats Little TheoryAccording to Liskov (221), Eulers Totient theorem is a simplification of Fermats little theorem and works with every n that are relatively prime to a. Fermats little theorem only works for a and p that are relatively prime.a p-1 1 (mod p)ora p a (mod p)where p itself is prime.It is, therefore, clear that this equation fits in the Eulers Totient theorem for every prime p, as indicated in p, where p is a prime and is given by p-1.Therefore, to prove Eulers theorem, it is vital to first prove Fermats little theorem.Proof to Fermats Little TheoremAs earlier indicated, the Fermats little theorem can be expressed as followsap a (mod p)taking two numbers a and p, that are relatively prime, where p is also prime.The set of a a, 2a, 3a, 4a, 5a(p-1)a(i) dig another set of number 1, 2, 3, 4, 5.(p-1a)(ii)If one decides to take the modulus for p, each segment of the set (i) will be appropriate to an item in the randomness set (ii). Therefore, there will be one on one correspondence between the two set s. This can be proven as lemma 1.Consequently, if one decides to take the product of the first set, that is a x 2a x 3a x 4a x 5a x . (p-1)a as well as the product of the import set as 1 x 2 x 3 x 4 x 5 (p-1). It is clear that both of these sets are congruous to one another that is, each component part in the first set matches another element in the second set (Liskov 221).Therefore, the two sets can be represented as followsa x 2a x 3a x 4a x 5a x . (p-1)a 1 x 2 x 3 x 4 x 5 (p-1) (mode p).If one takes out the factor a p-1 from the left-hand side (L.H.S), the resultant equation will be Ap-1 a x 2a x 3a x 4a x 5a x . (p-1)a 1 x 2 x 3 x 4 x 5 (p-1) (mode p).If the same equation is divided up by 1 x 2 x 3 x 4 x 5 (p-1) when p is prime, one will obtaina p a (mod p)ora p-1 1 (mod p)It should be clear that each element in the first set should correspond to another element in the second set (elements of the set are congruent). Even though this is not obvious at the first step, it can still be prove through trio logical steps as followsEach element in the first set should be congruent to one element in the second set this implies that none of the elements will be congruent to 0, as pand a are relatively prime.No two numbers in the first set can be labeled as ba or ca. If this is done, some elements in the first set can be the same as those in the second set. This would imply that two numbers are congruent to each other i.e. ba ca (mod p), which would mean that b c (mod p) which is not true mathematically, since both the element are divergent and less than p.An element in the first set can not be congruent to two numbers in the second set, since a number can only be congruent to numbers that differ by multiple of p.Through these three rules, one can prove Fermats Little Theorem.Proof of Eulers Totient TheoremSince the Fermats little theorem is a special form of Eulers Totient theorem, it follows that the two proofs provided earlier in this exploration a re similar, but slight adjustments need to be made to Fermats little theorem to justify Eulers Totient theorem (Krizek 97). This can be done by using the equationa n 1 (mod n)where the two numbers, a and n, are relatively prime, with the set of figures N, which are relatively prime to n 1, n1. n2.n n . This set is likely to drive home n element, which is defined by the number of the relatively prime number to n. In the same way, in the second set aN, each and every element is a product of a as well as an element of N a, an1, an2, an3ann.Each element of the set aN must be congruent to another element in the set N (mode n) as noted by the earlier three rules. Therefore, each element of the two sets will be congruent to each other (Giblin 111).In this scenario case, it can be said thata x an1 x an2 x an3 x . an n a x n1 x n2 x n3 x .n n (mod n).By factoring out a and an from the left-hand side, one can obtain the following equationa n 1 x n1 x n2 x n3 x .n n 1 x n1 x n2 x n3 x .n n (mod n)If this obtained equation is divided by 1 x n1 x n2 x n3 x .n n from both sides, all the elements in the two sets will be relatively prime. The obtained equation will be as followsa n 1 (mod n)Application of the Eulers TheoremUnlike other Eulers works in the number theory like the proof for the two-square theorem and the four-square theorem, the Eulers totient theorem has real applications across the globe. The Eulers totient theorem and Fermats little theorem are commonly used in decryption and encryption of data, especially in the RSA encryption systems, which protection resolves around big prime numbers (Wardlaw 97).ConclusionIn summary, this theorem may not be Eulers most well-designed piece of mathematics my favorite theorem is the two-square theorem by unnumbered descent. Despite this, the theorem seems to be a crucial and important piece of work, especially for that time. The number theory is still regarded as the most useful theory in mathematics nowadays. Thr ough this proof, I have had the opportunity to connect some of the work I have earlier done in discrete mathematics as well as sets relation and group options. Indeed, these two options seem to be among the purest sections of mathematics that I have ever studied in mathematics. However, this exploration has enabled me to explore the relationship between Eulers totient theorem and Fermats little theorem. I have also applied knowledge from one discipline to the other which has broadened my view of mathematics.Works CitedGiblin, P J. Primes, and Programming An Introduction to Number Theory with Computing. Cambridge UP, 1993. Print.Griffiths, H B, and Peter J. Hilton. A Comprehensive Textbook of Classical Mathematics A Contemporary Interpretation. London avant-garde Nostrand Reinhold Co, 1970. Print.Krizek, M., et al. 17 Lectures on Fermat Numbers From Number Theory to Geometry. Springer, 2001. Print.Liskov, Moses. Fermats Little Theorem. Encyclopedia of Cryptography and Security, pp. 221-221.Wardlaw, William P. Eulers Theorem for Polynomials. Ft. Belvoir Defense Technical Information Center, 1990. Print.

Just Mercy by Bryan Stevenson Review

Just Mercy by Bryan Stevenson check outBeacon Lights Book ReviewJust Mercy by Bryan StevensonEvery once in a while a hold up comes along that makes you think and even changes the way you meet previously looked upon a subject. Just Mercy is such a nurse. This is the true story of a young lawyers fight for arbiter on the part of more on death row. The author, Bryan Stevenson battles racial injustice and represents many poor, non-white, mentally ill and young offenders. Their stories are interspersed throughout the book as he recounts his defense of Walter McMillian, a black man on death row. The memoir reads like a legal thriller, juxtaposing his many triumphs and failures.Bryan Stevenson grew up poor in Delaware. His great-grandparents had been slaves in Virginia and this legacy of thrall influenced the way his grandparents raised their children and grandchildren. They particularly emphasized the importance of faith and education. Stevensons faith was cultivated in the Africa n Methodist Episcopal Church where he play the piano and sang in the choir. Stevenson attended Eastern College, a Christian institution outside Philadelphia, and then Harvard Law School. His focus to defend the poor began during college when he took an intensive class on race and poverty litigation. He was required to spend a month with an organization doing social justice work. They sent him to gallium to work with the Southern Prisoners Defense Committee where his first case required him to meet with a condemned man on death row.In the late 1980s, Stevensons legal firm first took up the cause of Walter McMillian, who was wrongfully convicted and sentenced to death for the murder of a white woman. The states case had many inconsistencies. They manufactured stories from witnesses who said they were with Walter when he perpetrate the crime and disregarded accounts from many eyewitnesses who said they were with Walter at a church fundraiser. The legal system was determined to find soulfulness to convict for this murder and discrete Walter would be prosecuted because of his affair with a prominent white woman (a crime during this time period). The exposition of this case opened my eyes to the unfortunate fact that since prosecutors and legal philosophy have legal immunity they can do considerable harm to innocent citizens when they are on the hunt for justice.A large portion of this book deals with the cases of poor black children sentenced to adult prisons and serving life sentences. Stevenson tells these stories very convincingly and sympathetically. He challenges their sentences because he says they are juveniles and his firm recognized the incongruity of not allowing children to smoke, drink, pose because of their lack of maturity and judgement while simultaneously treating some of the most at-risk, neglected, and impaired children exactly the same as full-grown adults in the criminal justice system. Circumstances that would bring a youth to be put in a situation where he felt his only option was to kill someone because he is not aware of his other options, should be recognized upon sentencing. One example is a young boy who shoots his mothers abuser. Yes, punishment needs to occur, but placing a young child in an adult prison is not always appropriate justice.Stevenson also defends many poor white women who have been convicted of the murder of their col crunchator or stillborn child. He tells the stories of pregnant women who are too poor to see a doctor or go to the hospital, and then unfortunately during labor deliver a stillborn child. These women are then arrested and charged with capital murder which is punishable by the death penalty.Not all of the characters in Stevensons book are sympathetic and his defense of them does not always seem to be justified. As a defense lawyer his heart may sometimes come in in the way of seeing their crimes objectively. In particular, he unsuccessfully defends a prisoner who commits the hein ous act of killing a child with a die meant for a neighbor.While justice has been denied for a large portion those in the prison system, the authors faith in both the power of redemption and justice, underlines his continued wish in the possibility of change. The true measure of our character, Stevenson writes, is how we treat the poor, the disfavored, the accused, the incarcerated, and the condemned we all need mercy, we all need justice, and perhaps we all need some measure of gratuitous grace.

Composting and the Benefits and Limitations of its Use as Soil Amendmen

Composting and the Benefits and Limitations of its Use as Soil Amendment Composting is widely-known as an environmentally sustainable method of recycling food scraps and garden/yard clippings. jibe to the United States Environment Protection Agencys (EPA) 1994 report entitled, devour Prevention, Recycling, and Composting, food discards constitute approximately 8 percent of municipal solid waste generated nationwide. A character study observed in San Francisco, California, reported that an estimated 31 percent of residential wastes and 19 percent of commercial waste generated is food waste. Composting is a viable behave to the food-waste problem. Composting not only reduces the count of waste, buy also contains chemical properties and other rich elements that benefit the soils quality, allowing farmers to use it as soil amendment. The following research bequeath examine the process of composting and its different variables in order to investigate its positive and negative affect s on the soil. Composting refers to the controlled decomposing of organic matter by microorganisms, mainly bacteria and fungi, that dishonour organic matter down into readily available nutrients. In truth, it relies on little to no human intervention which makes it fairly easy. There are literally dozens of different technologies that are currently used to create compost (Stickelberger, 1975, p.188). Professor David Bice of Carleton College utilizes a method known as vermicomposting. Dictated by the amount of space (he is limited to in his apartment), Professor Bice places his food scraps in a bucket where red-colored worms digest and convert the bio-waste into a fertilizer product. Therefore, selection of a system depends largely on pragmatic factors.... ...lized form of organic matter that improves the physical, chemical, and biological properties of soil.Works CitedStickelberger, D, 1975, Survery of city refuse composting. In Organic Materials and Fertilizers Rome, Italy, FAO S oils, Bulletin 27.Kapage, F.S.C.P, 1974, tropical Soils Classification, Fertility, and Management New York, St. Martin Press, 201p.Broadbent, F.E., 1987, Organic Matter . Minnesota Compost and Co-Compost Research Project, In USDA Yearbook of Agriculture U.S. Government Printing Office, Washington, D.C.Government Document, February 1994, Waste Prevention, Recycling, and Composting Options Lessons From 30 Communities U.S. Environmental Protection Agency, Washington D.C. 47p.Government Document, May 1994, Composting Yard Trimmings and Municipal Solid Waste U.S. Environmental Protection Agency, Washington D.C. 56-57p.

journalism Essay -- essays research papers

In my twenty years and some months existence in this world, Ive been fortunate enough to personify a secured disembodied spirit. I believe nutrient and shelter are two of the most important essentials in life, and Ive never had to worry about food not being on the table or if I had a place to sleep at night which is why I consider myself to vex a secure life. Volunteering at Simpson Housing helped me realize many things in life, and also made me wonder why some people are so fortunate in life while others struggle to survive in this world.Prior to Simpson Housing, I was vaguely familiar with the issue of homelessness. I only knew the al-Qaida of it, but never fully understood it. There are a lot of misconceptions and stereotypes about the homeless and Ive learned that you cant really judge until youve experienced it. You hear a lot of mischievously things about homeless people- theyre lazy, drug addicts, bums, freeloaders, etc Its easy to focus on the bad because you remember the bad more than the good. It must be hard for homeless people because they are pre-judged without even getting to be known. They get rejected in advance getting a chance. I will admit that I did believe some of the misconceptions of homeless people because I had atomic number 53 negative experience. When I was younger, my family and I utilize to go eat at Rainbow, a Chinese restaurant, in Minneapolis. We always went at night and on the way home, there was a disrobe where all the homeless people would camp out. I couldnt believe people had to live wish that. Ive had several encounters with homeless people where theyd come up to me asking for money until this once incident. I was with my dad and this lady and her son approached us asking for some help. I had always prone a dollar or two before, but I saw she had a son so I convinced my dad to spend them atomic number 23 dollars. They seemed to be really gracious of our donation and we felt really good about helping them. Fi ve minutes later, we saw the lady and her son turn on away in a brand new Toyota Camry. My dad and I were completely stunned. I was so angry. There are so many others that couldve used the money we had given. The lady took advantage of us, and completely had no morals whatsoever. However, that did not stop me from wanting to give help to the needy. Instead of giving them money, I give them food. I know its not exactly that same, but I still do feel that I am helping them. ... ... but I certain some people have no control over it and that really saddens me. My heart goes out to homeless children because they are so young. They have so much more to worry about when they should only be focusing on having fun as a child and getting their education. However quite of being able to focus on school, they have to focus on surviving. Life isnt fair, but then again what is in life? Situations like homelessness is not something many can touch base with, but it is something we can all take p art in to help reduce it. No one deserves to live a life where nothing at all is certain. With so many mistaken beliefs on homeless people, those who are fortunate in life need to realize that those who are homeless are strong in heart. They dont give up and they keep living each solar day in hope for a better tomorrow. Ive learned through volunteering at Simpson Housing is that you have to take the good with the bad and learn from it all. get dressedt let the bad hold you down and you should embrace it and turn it into something positive. Whatever may come your way, you should appreciate it because it only helps you grow into the person you are to become.

Divisive, but Effectual: Reinstating the Draft Essay -- U.S. Military

Ever since World War II, the last war that garnered full American support, the percentage of American citizens enrolling in the U.S. Army has attenuated significantly. The U.S. is a major military power in the present world American soldiers atomic number 18 engaged in many multi-theater positions, consequently dispersing troops oer various regions worldwide. As only one in ten citizens has served in the US Army, American knowledge of military conditions is rapidly lessening along with the national harmony that complements such knowledge. Therefore, the comprehension that American security is closely associated with a formidable military never resonated within the greater part of modern American citizens. Although antiwar activists may decry compulsory conscription as unconstitutional, it needs to be reinstated in order to solidify the meaning of being an American citizen, boost the parsimoniousness by providing jobs, and unify the American hatful with enhanced homeland security .Generally, reinstating the Constitutional draft would solidify the meaning of being an American citizen. Often, wealthier and more privileged people are able to ask for exemption from serving, so representation of soldiers in combat is composed of less-privileged citizens. With compulsory conscription, all American citizens, men or women, ages 18-26 are subject to be recruited for the armed forces. In fact, only 4 members of the 107th Congress who voted in favor of the Iraq war had children serving there. Politicians would know how it feels to have loved ones in a war, instead of freely speaking about wartime without experiencing it in any way. 1 (206) Americans feel that public support for war in foreign countries is not strong because the soldiers fighting are poorly rep... ... CQ Researcher Outline, January 11, 1991. http//library.cqpress.com/cqresearcher/cqresrre1991011100 (accessed March 19, 2012).Binding the Nation National Service in America. U.S. Army War College, 2008. ht tp//www.carlisle.army.mil/usawc/parameters/Articles/08winter/grigsby.pdf (accessed March 12, 2012). bring out about National Service. Americans for a National Service Act, . nationalserviceact.org (accessed March 19, 2012). Greenblatt, Alan. Upward Mobility. CQ Researcher, April 29, 2005. http//www.cqpress.com/product/Researcher-Upward-Mobility-v15-16.html (accessed March 14, 2012).Farrell, Brenda S. Military Personnel Evaluation Methods Linked to Anticipated Outcomes take to Inform Decisions on Army Recruitment Incentives. U.S. Government Accountability Office, September 19, 2008. http//www.gao.gov/products/GAO-08-1037R (accessed March 19, 2012).

The Devil in Joyce Carol Oates Where Are You Going, Where Have You Bee

The Devil in Joyce Carol Oates Where argon You Going, Where Have You Been? Her name is Connie, and she is not unlike many girls of the period she lives in. She is vain, she is constantly at war with her family, and she is in an incredible rush to grow up. Her race to maturity is the trait focused on in Joyce Carol Oates Where Are You Going, Where Have You Been. It splits Connie into two different personalities One for home, and one for anywhere that was not home (431). Everything astir(predicate) her?her walk, her smile, and her laugh?metamorphoses as soon as she steps out the front door. The baby is hidden, the seductive young woman emerges, and the world of the ?big kids? is more than willing to take her in. This world is what she thinks she wants, until the day a shiny golden convertible pulls into her driveway and the the mysterious Arnold Friend emerges. Through Arnold Friend, Connie learns that her rush to grow up is foolish and that she is trying to jump in to a world that she knows nothing about and that could be potentially dangerous. She ultimately releases her dream and clings to her family as never before, realizing that their firm grasp on her is not for their benefit, but her own. Joyce Carol Oates?s vivid description of Arnold Friend carries the around emotional freight, as the evil behind his apparent glamor brings about Connie?s change. Though he takes the outer appearance of a normal boy, everything about his behavior suggests that he is the Devil himself in disguise. The most obvious aspects of Arnold Friend that suggest that he is the Devil in disguise are his physical features. For example, several(prenominal) references are made to the abnormality of his feet. As he walks about, he stumbles... ...s images of Arnold Friend, suggesting that he is the Devil in disguise, make the story both moving and successful. Connie?s dread at the face of the Devil is shared by Oates?s audience, making it easier for them to u nderstand both the emotional impact of her struggle and the gravity of the lesson being taught by the story things are not always as they seem. Connie?s fright at the discovery of Arnold?s true nature is comparable to Eve?s fright at the awareness of good and evil. Connie jumped into a world that was as appealing to her as the serpent?s shiny red apple was to Eve, only if to discover that everything beyond the savory exterior was fraught with venomous poison. Reference Oates, Joyce Carol, ?Where Are You Going, Where Have You Been?? reprinted in X.J. Kennedy?s Literature An introduction to Fiction, Poetry, and Drama, 5th ed (Harper Collins, 1991).

Jonsons Volpone :: Jonson Volpone

 Homosexual Overtones in Volpone    During the metempsychosis, women did not participate in the theatre hence, men, dressed in drag, played womens percentages. This particular characteristic of Renaissance drama adds many dimensions, erotic and social, to the spectacle on the stage. However, The primary difficulty in surveying this landscape results from the strong indications that early modern eroticism was fundamentally divergent from that today. Consequently, the challenge deciphering what may be radically different cultural codes for the Rena issance is formidable (Zimmerman 7). The interactions between cross-dressed boy actors and the adult young-begetting(prenominal) actors, by todays standards, would be considered homo-erotic. In Ben Johnsons Volpone, the role of Celia, the main fe mannish character, would have be en played by a cross-dressed boy hence, many inferences about Renaissance eroticism may be made by exploring the element of cross-dressing and h ow it transforms the action on stage and the audiences perception. Celia (played by a cross-dressed boy) interacts with men throughout the play. The scene in which Volpone attempts to rape Celia could be construed as extremely homo-erotic. Volpone desires Celia, yet she refuses to succumb to his advances thus he trie s to force himself upon her. Bonario is Celias true love interest, which also has homosexual overtones. The sexual and intimate interactions between Celia and the manful characters creates an interesting dynamic. For instance, When an actor in a male role did not need to impersonate adult-ness, his interaction with a cross-dressed actor, particularly a cross-dressed boy, change. Presumably, the adult actor, by virtue of age, voice, physical appearance and interpretive range, lent credence to the (usually) heterosexual valences of cross-dressing within that fiction. The dual lens on the dramatic action that the adult male actor provided was in all likelihood angled most directly at adult male spectators (Zimmerman 46). The actors are interacting like hete rosexuals of the opposite sex, however, the fact that it is really two males blurs the lines of what the audience was actually seeing and enjoying. Celia was obviously made to be attractive to the male spectators, because she is the main female love interest in the play. The male spectators may be attracted to what aesthetically appears to be a woman, or they may be attracted to an androgynous, cr oss-dressed boy. Hence, Celias appeal is twofold.