Search results
Results From The WOW.Com Content Network
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 ...
Main page; Contents; Current events; Random article; About Wikipedia; Contact us; Pages for logged out editors learn more
Therefore, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for n = 4 and for all odd primes p. For any such odd exponent p , every positive-integer solution of the equation a p + b p = c p corresponds to a general integer solution to the equation a p + b p + c p = 0 .
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]
For a positive integer a, if a composite integer x divides a x−1 − 1, then x is called a Fermat pseudoprime to base a. [ 1 ] : Def. 3.32 In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a . [ 2 ]
Shanks' square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success of Fermat's method depends on finding integers x {\displaystyle x} and y {\displaystyle y} such that x 2 − y 2 = N {\displaystyle x^{2}-y^{2}=N} , where N {\displaystyle N} is the ...
There are infinitely many Fermat pseudoprimes to any given basis a > 1. [1]: Theorem 1 Even worse, there are infinitely many Carmichael numbers. [2] These are numbers for which all values of with (,) = are Fermat liars. For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors.
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.. A Gaussian integer is a complex number + such that a and b are integers. The norm (+) = + of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.