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Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema.
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 ...
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 ...
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. [1]
For example, 1093 2 = 1194649 is a Fermat pseudoprime to base 2, and 11 2 = 121 is a Fermat pseudoprime to base 3. The number of the values of b for n are (For n prime, the number of the values of b must be n − 1, since all b satisfy the Fermat little 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.
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)).
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.