For over 350 years, proving fermats last theorem was the most notorious unsolved mathematical problem, a puzzle whose basics. Fermats last theorem simple english wikipedia, the free. Fermats little theorem states that if p is a prime number, then for any integer a, the number a p a is an integer multiple of p here p is a prime number a p. There is some doubt about this for various reasons. In this paper we show an alternative perspective on fermat s last theorem using notions of classical geometry, trigonometry, reductio ad absurdum, and simple but nonobvious mathematical tricks. There are deep and subtle connections between number theory. In the centuries following the initial statement of the result and its general proof, various proofs were devised for particular values of the exponent n. Type in any equation to get the solution, steps and graph. But the most surprising thing is that fermats last theorem is a special case of the.
Any such elliptic curve has the property that its hasselveil zeta function has an analytic continuation and satisfies a functional equation of the standard type. Chapter 8 fermats little theorem trinity college dublin. How did fermat prove his last theorem for mathn4math. It seems to be the only direct proof currently existing. Fermats last theorem is one of the most beguiling results in mathematics. Fermats last theorem is a more general form of the equation. Pdf i give a relativistic proof to the fermat s last theorem. The fermat factorization method revisited robert erra. If you look at the theorem you can see why fermat might have thought that he found an elegant proof. Let p be a prime which does not divide the integer a, then a p 1. Fermat s last theorem by wiles and taylor there was that of the italian mathematician onofrio gallo b. It was 50 minutes of mathematicians talking about mathematics, which is not the obvious recipe for a tv blockbuster, but the result was a programme that captured the public imagination and which received critical acclaim. It was found written in the margins of fermats copy of arithmetica, an ancient number theory text by diophantus. Christophe grenier 30th june 2009 abstract we consider the well known fermat factorization method, we call the fermat factorization equation the equation solved by it.
Fermat s little theorem is considered a special case of eulers general totient theorem as fermat s deals solely with prime moduli, while eulers applies to any number so long as they are relatively prime to one another bogomolny, 2000. Given any integer solution, can produce a smaller integer solution. If p is a prime number and a is any other natural number not divisible by p, then the number is divisible by p. If a is not divisible by p, fermats little theorem is equivalent to the statement that a p1 1 is an integer multiple of p. In 1996, working with john lynch, i made a documentary about fermats last theorem for the bbc series horizon. Thanks for contributing an answer to mathematics stack exchange.
Background and history of fermats little theorem fermats little theorem is stated as follows. For over 350 years, proving fermats last theorem was the most notorious unsolved mathematical problem, a puzzle whose basics most children could grasp but whose solution eluded the greatest minds in the world. If we dont care about which way each senator voted, the answer is 2n2 2n 1. In this paper we show an alternative perspective on fermats last theorem using notions of classical geometry, trigonometry, reductio ad absurdum, and simple but nonobvious mathematical tricks. The number 2 is not divisible by the prime 11, so 210. There are several generalizations of the fermat equation to more general equations that allow the exponent n to be a negative integer or rational, or to consider three different exponents. Fermat s theorem stationary points, about local maxima and minima of differentiable functions. If a is not divisible by p, fermats little theorem is equivalent to the statement that a p11 is an integer multiple of p. Theorem fermat the only pairs of rational numbers fractions x and y that satisfy the equation y2 x3 x are 0.
Consider the set of integers ill show that they reduce mod p to the standard system of residues, then apply wilsons theorem. Fermats last theorem by wiles and taylor there was that of the italian mathematician onofrio gallo b. Fermat polygonal number theorem, about expressing integers as a sum of polygonal numbers. Nigel boston university of wisconsin madison the proof of. Pythagorean triples, fermat descent, lecture 23 notes. If we try all the values from x 1 through x 10, we nd that 53 4 mod 11. The result is called fermats little theorem in order to distinguish it from. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. It is the purpose of this paper to provide such a proof. The shimurataniyama conjecture is part of a more general philosophy. Since c 2 c3 3 is the only cyclic decomposition of abelian groups of order 54 for which all elements have order dividing 6, it is su cient to show 6 1 mod 1.
The statement of the theorem involves an integer exponent n larger than 2. Proof of fermats last theorem for specific exponents. Sebbene il suo lavoro di magistrato fosse molto impegnativo, fermat trovo il tempo per dedicarsi in modo molto proficuo allo studio della matematica. The title fermats little theorem is sometimes given to the following variant. Fermats little theorem is a fundamental theorem in elementary number theory, which helps compute powers of integers modulo prime numbers. The proof of the fermats last theorem will be derived utilizing such a geometrical representation of integer numbers raised to an integer power. New proof of fermats little theorem the proof that follows relies on taylors theorem or the binomial theorem.
But the most surprising thing is that fermat s last theorem is a special case of the. It is called the little theorem to distinguish it from fermat s last theorem. The following corollary is, in fact, equivalent to fermats little theorem. Both fermats last theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous. What follows are two more direct proofs of fermats. Fermat s principle, about the path taken by a ray of light. Fermat s last theorem considers solutions to the fermat equation. The thesis of the demonstration is based on the assumption that the equation has at least one correct solution for arbitrary triple positive integers raised to the power n when n 2. A special case is when a, b and c are whole numbers. If p is a prime and a is any integer not divisible by p, then a p. Fermat s little theorem is the basis for the fermat primality test and is one of the fundamental results of elementary number theory.
Together with ribets theorem, it provides a proof for fermats last theorem. The leading thought throughout the derivation is illustrated in fig. Any such elliptic curve has the property that its hasselveil zeta function has an analytic continuation and satisfies a. It is a special case of eulers theorem, and is important in applications of elementary number theory, including primality testing and publickey cryptography. If p xthe result follows on multiplying the congruence in the theorem by x. The most important theorem is fermats little theorem which is the basis of the rsa.
Since c 2 c3 3 is the only cyclic decomposition of abelian groups of order 54 for which all elements have order dividing 6, it is su cient to show 6 1. Nigel boston university of wisconsin madison the proof. Proof of fermats last theorem for specific exponents wikipedia. Fermats little theorem states that if p is a prime number, then for any integer a, the number a p a is an integer multiple of p. But avoid asking for help, clarification, or responding to other answers. However, some people state fermats little theorem as. It is so easy to calculate a p 1 quickly modulo p that most elementary primality tests are built using a version of fermats little theorem rather than wilsons theorem. Sep 27, 2015 by fermats little theorem, 36 1 mod 7. Free equations calculator solve linear, quadratic, polynomial, radical, exponential and logarithmic equations with all the steps.
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