
Glossary: Prime Pages: Top 5000: 
GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) Fermat's method of factoring consists of finding x and y such that x^{2}y^{2}=n. The right side of the equation factors into (xy)(x+y), and if xy is not one, then you have found a nontrivial factorization. There exists several easy extensions to this idea. For example, instead of solving x^{2}y^{2}=n, we try to find x and y such that x^{2}=y^{2} (mod n). This means n divides x^{2}y^{2}, so if x is not congruent to +y modulo n, then gcd(xy,n) or gcd(x+y,n) must be a nontrivial factor of n. For example, suppose we wish to factor n=91. The first few squares (modulo 91) are as follows: 1, 4, 9, 16, 25, 36, 49, 64, 81, 9, 30, 53, 78, ...So we see 3^{2}=10^{2} (mod 91), and expect either gcd(103,91)=7 or gcd(10+3,91)=13 to be a proper divisor of 91. Both are! Fermat's method (and the simple extentions to it above) are not very efficient in finding factors themselves, but are theoretically important in that many more modern methods such as: the quadratic sieve, multiple polynomial quadratic sieve, and the special and the general number field sieves are all based upon this method. Contributed by Lucas Wiman
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Chris K. Caldwell © 19992019 (all rights reserved)
