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Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...
Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on Euler's theorem. We want to show that m ed ≡ m (mod n), where n = pq is a product of two different prime numbers, and e and d are positive integers satisfying ed ≡ 1 (mod φ(n)).
Gerhard Frey (German:; born 1 June 1944) is a German mathematician, known for his work in number theory.Following an original idea of Hellegouarch, [1] he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a purported solution to the Fermat equation, that is central to Wiles's proof of Fermat's Last Theorem.
Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation a n + b n = c n for any integer value of n greater than 2. (For n equal to 1, the equation is a linear equation and has a solution for every possible a and b.
Fermat's little theorem states that if p is prime and a is coprime to p, then a p−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides a x−1 − 1, then x is called a Fermat pseudoprime to base a. It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a. Some sources use variations of this ...
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. [1] One attempt by Germain to prove Fermat’s Last Theorem was to let p be a prime number of the form 8k + 7 and to let n = p – 1. In this case, + = is unsolvable. Germain’s proof, however, remained ...
Some of the proofs of Fermat's little theorem given below depend on two simplifications. The first is that we may assume that a is in the range 0 ≤ a ≤ p − 1 . This is a simple consequence of the laws of modular arithmetic ; we are simply saying that we may first reduce a modulo p .