Eulers Totient Theorem

Eulers Totient Theorem Summary   Ã‚   Euler Totient theorem is a generalized form of Fermats Little theory. As such, it solely depends on Fermats Little Theorem as indicated in Eulers study 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, but in a unique way. In this manner, The Eulers Totient function à Ã¢â‚¬   for any integer (n) can be demarcated, as the figure of positive integers is not greater than and co-prime to n. aà Ã¢â‚¬  (n) = 1 (mod n) Based 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. Based on this fact, some scholarly source refers to this theorem as the Fermats-Euler theorem of Eulers generalization. Introduction I first developed an interest in Euler when I was completing a listener 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 mathematics discipline, as he was responsible for proving numerous problems and conjectures. Taking the number theory as an example, Euler successively played a vital role in proving the two-square theorem as well as the Fermats little 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 many; it is a generalization of Fermats little theorem that is commonly known as Eulers theorem. Theorem Eulers Totient theorem holds that if a and n are coprime positive integers, then since ÃŽÂ ¦n is a Eulers Totient function. Eulers Totient Function Eulers Totient Function (ÃŽÂ ¦n) is the count of positive integers that are less that n and relatively prime to n. For instance, ÃŽÂ ¦10 is 4, since there are four integers, which are less than 10 and are relatively prime to 10: 1, 3, 7, 9. Consequently, ÃŽÂ ¦11 is 10, since there 11 prime numbers which are less than 10 and are relatively prime to 10. The same way, ÃŽÂ ¦6 is 2 as 1 and 5 are relatively prime to 6, but 2, 3, and 4 are not. The following table represents the totients of numbers up to twenty. N ÃŽÂ ¦n 2 1 3 2 4 2 5 4 6 2 7 6 8 4 9 6 10 4 11 10 12 4 13 12 14 6 15 8 16 8 17 16 18 6 19 18 20 8 Some 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 ÃŽÂ ¦10 = 4. Then this relation can also be represented as follows: 34 = 81 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mode 10). Conversely, if n = 15 and a = 2, it is clear that 28 = 256 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod 15). Fermats Little Theory According 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 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod p) or a p à ¢Ã¢â‚¬ °Ã‚ ¡ 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 ÃŽÂ ¦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 Theorem As earlier indicated, the Fermats little theorem can be expressed as follows: ap à ¢Ã¢â‚¬ °Ã‚ ¡ 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, 5aà ¢Ã¢â€š ¬Ã‚ ¦(p-1)a}à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦(i) Consider another set of number {1, 2, 3, 4, 5à ¢Ã¢â€š ¬Ã‚ ¦.(p-1a)}à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦(ii) If one decides to take the modulus for p, each element of the set (i) will be congruent to an item in the second set (ii). Therefore, there will be one on one correspondence between the two sets. 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 à ¢Ã¢â€š ¬Ã‚ ¦. (p-1)a } as well as the product of the second set as {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (p-1)}. It is clear that both of these sets are congruent to one another; that is, each element in the first set matches another element in the second set (Liskov 221). Therefore, the two sets can be represented as follows: {a x 2a x 3a x 4a x 5a x à ¢Ã¢â€š ¬Ã‚ ¦. (p-1)a } à ¢Ã¢â‚¬ °Ã‚ ¡ {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (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 à ¢Ã¢â€š ¬Ã‚ ¦. (p-1)a } à ¢Ã¢â‚¬ °Ã‚ ¡ {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (p-1)} (mode p). If the same equation is divided by {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (p-1)} when p is prime, one will obtain a p à ¢Ã¢â‚¬ °Ã‚ ¡ a (mod p) or a p-1 à ¢Ã¢â‚¬ °Ã‚ ¡ 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 proved through three logical steps as follows: Each 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 à ¢Ã¢â‚¬ °Ã‚ ¡ ca (mod p), which would mean that b à ¢Ã¢â‚¬ °Ã‚ ¡ 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 Theorem Since the Fermats little theorem is a special form of Eulers Totient theorem, it follows that the two proofs provided earlier in this exploration are similar, but slight adjustments need to be made to Fermats little theorem to justify Eulers Totient theorem (KrÃÅ'Å’iÃÅ' zÃÅ'Å’ek 97). This can be done by using the equation a ÃŽÂ ¦n à ¢Ã¢â‚¬ °Ã‚ ¡ 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. n2à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n }. This set is likely to have ÃŽÂ ¦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, an3à ¢Ã¢â€š ¬Ã‚ ¦anÃŽÂ ¦n}. 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 that: {a x an1 x an2 x an3 x à ¢Ã¢â€š ¬Ã‚ ¦. an ÃŽÂ ¦n } à ¢Ã¢â‚¬ °Ã‚ ¡ {a x   n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n } (mod n). By factoring out a and aÃŽÂ ¦n from the left-hand side, one can obtain the following equation a ÃŽÂ ¦n {1 x n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n} à ¢Ã¢â‚¬ °Ã‚ ¡ {1 x n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n } (mod n) If this obtained equation is divided by {1 x n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n } from both sides, all the elements in the two sets will be relatively prime. The obtained equation will be as follows: a ÃŽÂ ¦n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n) Application of the Eulers Theorem Unlike 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). Conclusion In summary, this theorem may not be Eulers most well-designed piece of mathematics; my favorite theorem is the two-square theorem by infinite 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. Through 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 Cited Giblin, 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: Van Nostrand Reinhold Co, 1970. Print. KrÃÅ'Å’iÃÅ' zÃÅ'Å’ek, 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.

Beattie noted in his history paper Essay

Shakespeare died over 400 years ago. So, Hiram inserted a quote by Shakespeare from the play Hamlet into an essay without using quotation marks or citing the reference. I disagree that the essay was not cited and no questions  marks were used when the quote was inserted. If you do not cite a quote that was copied from a text, even if the author has died, this was being considered plagiarism. If you would have paraphrased the quote that would be difference because you are taking pieces of the quote and putting the rest in your own words. Beattie noted in his history paper the fact that representatives from the 13 British colonies signed the United States Declaration of Independence in 1776, but he did not cite a source for the information. I disagree that Beattie did not cite the source. Consider that he noted a fact that the British colonies signed the Declaration in the year 1776. He mentions the year of when the Declaration was signed, so this should be cited. When taking notes about the American Revolution. Sara found a quote of Benjamin Franklin’s words, which she paraphrased. She inserted it into her paper without quotation marks or a reference to Benjamin Franklin. I disagree that Sara did not cite her paper. When paraphrasing if you state the author and the year, a source should be cited. Jeffery wrote a paper on popular music in which he claimed that many songs came from earlier sources. He included the entire song lyrics from the Beatles and compared them to the lyrics of a song by a folk singer from the 1930s. He cited the Beatles. Although he included the lyrics from the 1930 song, he did not include a reference to the source. I agree that Jeffrey cited the singer of the song lyrics. I disagree that he did not reference the source. Whenever you site a source from a textbook you should include a reference page. The reference gives you all the information on where you have found your information.

Jeanne Wilson Essay

Jeanne Wilson was employed as a nurse at the Mary McClellan hospital in the late 1970s. While employed there she heard tale of â€Å"The Pink Lady† who roamed the halls of the maternity ward and seemed drawn to a certain room, with a particular patient. â€Å"Well, when I was working at the Mary McClellan Hospital in Cambridge (that’s just across the river from here) I had the night watch of the maternity ward. The other nurses used to tell me to watch myself up there because of this pink lady. I most certainly believe in ghosts although I have never seen any myself you understand. I was never really afraid, I just figured she’d be peaceful and never bother any of us doing the good work up on the floor. Sometimes at night you would hear the swoosh, swoosh of someone’s bathrobe rubbing on their legs or the scuffle sound of slippers on the floor. There would never be anyone there, but you’d hear it just the same. This one time we had a lady come in who was dying. She was an older woman and the nurses really loved her. Well, she asked to die on the maternity ward so all of the nurses felt this was ok. We brought her up and got her settled. It was a very small hospital you know. Anyway, I was checking in patients late at night and as I was walking past her room, I noticed the door was shut. I looking in through the little window on those doors and there was the pink lady! She had pink curlers in her hair, a pink robe on, and pink slip-on kind of slippers on her feet! Well, I almost died myself right there. I couldn’t believe my eyes. When I caught my breath, I peeked back in the door to see her just standing over the old woman just watching her sleep. I creaked open the door and sure enough she disappeared. They said the pink lady only walked the maternity ward because she had lost all of her children in those rooms. When she died of cancer later on, she had asked to be in the maternity ward to die like her children! Well, I never expected her to visit this woman. Of course, when I think of it now, it makes perfect sense! They don’t deliver babies in the hospital anymore – actually its not even a hospital now. But, that’s the pink lady and I’m telling you – she’s real. † Ms. Wilson states this story as a full-fledged memory from her past. While employed at the hospital she points out others telling her this singular truth, and denied knowledge of any other tales of this nature in the ward. The story comes from a region other than place interviewed, not allowing for cross-reference with other individuals. Hospital stories of this nature are found throughout history, with a higher concentration surrounding Civil War hospitals. The time-frame for the inception of the tale can best be ascertained as the 1950s era due to the physical description of the pink lady. One can assume the story originated at that time. The telling was very excited with strong body language used throughout. Exaggerated facial expressions and multiple hand manipulations of air demonstrated actual behavior during the encounter. The skills of her trade do not enter the story, nor are they necessary other than putting Ms. Wilson in the hospital after hours. Ms. Wilson was interviewed with her elderly mother present. This parent did not obviously believe the tale, nor did she appreciate the telling of it. This did not hinder the younger Wilson woman; in fact, she seemed eager for acceptance of the tale. The recording of her story may well have promoted a more fascinating telling of the story with added bits of detail. The education level of Ms. Wilson was undetermined. Apparently, she is not a qualified nurse, having never gone to school to keep up with the educational demands of the field. She currently works in a rest home. The idea of her having been a nurse at the time appeared to have given the tale more validity in her mind as she felt it was a scientific study of sorts. She was reminded of the nature of the recording. References Wilson, J. (personal communication, October 14, 2006)

Treatment And Prognosis For Schizophrenia - 1261 Words

Mansi Patel Intro to Psychology Due: April 28, 2016 Treatment and Prognosis For Schizophrenia Schizophrenia is a brain disorder that cannot be cured but there are various types of treatments that are available for lifetime support. Some of which are, Coordinated Specialty Care treatment which is aims at improving the quality of life by giving psychosocial therapies, family involvement, and education support. These specialists will give a certain type of treatment plan to the patients for them to follow and work together to make therapy decisions. Psychotherapy, and this treatment is a way to help patients understand their illness and manage their symptoms better. There are also different types of psychotherapy called Cognitive Behavioral†¦show more content†¦Usually taking antipsychotic medications are recommended for improvement. To be more broad, people with this disorder are also exposed to various treatment plans as well. Although this illness includes many medications, but the most effective one is antipsychotic medications- A commonly used drug that treats schi zophrenia; which are tablets that control psychotic symptoms such as delusions, hallucinations, or other symptoms that the patient feels when having this disorder. Using this drug there are also side effects such as dry mouth, drowsiness, muscle stiffness, and tardive dyskinesia. If a person has used antipsychotic medication for long periods of time, eventually they might increase their risk of developing a movement disorder called tardive dyskinesia, briefly talked above which cannot be cured and can result in the malfunction of moving their face and jaws. Note that this medication reduces the symptoms of schizophrenia, not take away the cause for it. Along with the brain, antipsychotic drugs reduce the syndrome of psychosis as well as reducing the symptoms of bipolar disorder or a manic depression which are alternating periods of mood swings that lead to depression. The way this medication works is it can either decrease or alter the effect of neurotransmitters in the brain. Neuro transmitters helps transfer messages throughout the brain and along with that, dopamine and serotonin