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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 ...
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes. A Gaussian integer is a complex number a + i b {\displaystyle a+ib} such that a and b are integers. The norm N ( a + i b ) = a 2 + b 2 {\displaystyle N(a+ib)=a^{2}+b^{2}} of a Gaussian integer is an integer equal to the square of the absolute value ...
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem. [3]
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]
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Sir Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using previous ...
Near the beginning of the 20th century, it was shown that a corollary of Fermat's little theorem could be used to test for primality. [8] This resulted in the Pocklington primality test . [ 9 ] However, as this test requires a partial factorization of n − 1 the running time was still quite slow in the worst case.
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 .