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Mirimanoff's Congruence
In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are interesting in their own right. The theorem is due to Dmitry Mirimanoff. Definition The ''n''th Mirimanoff polynomial for the prime ''p'' is :\phi_n(t) = 1^t + 2^t^2 + ... + (p-1)^ t^. In terms of these polynomials, if ''t'' is one of the six values where ''X''''p''+''Y''''p''+''Z''''p''=0 is a solution to Fermat's Last Theorem, then * φ''p''-1(''t'') ≡ 0 (mod ''p'') * φ''p''-2(''t'')φ2(''t'') ≡ 0 (mod ''p'') * φ''p''-3(''t'')φ3(''t'') ≡ 0 (mod ''p'') :... * φ(''p''+1)/2(''t'')φ(''p''-1)/2(''t'') ≡ 0 (mod ''p'') Other congruences Mirimanoff also proved the following: *If an odd prime ''p'' does not divide one of the numerators of the Bernoulli number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20 The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of ''Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, Wiles's proof of Fermat's Last Theorem, the first success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dmitry Mirimanoff
Dmitry Semionovitch Mirimanoff (; 13 September 1861, Pereslavl-Zalessky, Russia – 5 January 1945, Geneva, Switzerland) was a member of the Moscow Mathematical Society in 1897. And later became a doctor of mathematical sciences in 1900, in Geneva, and taught at the universities of Geneva and Lausanne. Mirimanoff made notable contributions to axiomatic set theory and to number theory (relating specifically to Fermat's Last Theorem, on which he corresponded with Albert Einstein before the First World WarJean A. Mirimanoff. Private correspondence with Anton Lokhmotov. (2009)). In 1917, he introduced, though not as explicitly as John von Neumann later, the cumulative hierarchy of sets and the notion of von Neumann ordinals; although he introduced a notion of regular (and well-founded set) he did not consider regularity as an axiom, but also explored what is now called non-well-founded set theory and had an emergent idea of what is now called bisimulation. Life Dmitry Semion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wieferich Prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the ''abc'' conjecture. , the only known Wieferich primes are 1093 and 3511 . Equivalent definitions The stronge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the '' American Journal of Mathematics''. At his death, he was a professor at Oxford University. Biography James Joseph was born in London on 3 September 1814, the son of Abraham Joseph, a Jewish merchant. James later adopted the surname ''Sylvester'' when his older brother did so upon emigration to the United States. At the age of 14, Sylvester was a student of Augustus De Morgan at the University of London (now University College London). His family withdrew him from the university after he was accused of stabbing a fellow student with a knife. Subsequently, he attended the Liverpool Royal Institutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paulo Ribenboim
Paulo Ribenboim (born March 13, 1928) is a Brazilian-Canadian mathematician who specializes in number theory. Biography Ribenboim was born into a Jewish family in Recife, Brazil. He received his BSc in mathematics from the University of São Paulo in 1948, and won a fellowship to study with Jean Dieudonné in France at the University of Nancy in the early 1950s, where he became a close friend of Alexander Grothendieck. He has contributed to the theory of ideals and of valuations. Ribenboim has authored 246 publications including 13 books. He has been at Queen's University in Kingston, Ontario, since the 1960s, where he remains a professor emeritus. Jean Dieudonné was one of his doctoral advisors. Andrew Granville, Jan Minac, Karl Dilcher and Aron Simis have been doctoral students of Ribenboim. The Ribenboim Prize of the Canadian Number Theory Association is named in his honor. Personal life In 1951, Ribenboim married Huguette Demangelle, a French Catholic woman wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
