2 = , and the factors are roughly equal: Fermat's, applied to Nuv, will find them quickly. ≈ N O ) The end result is the same: a difference of square mod n that, if nontrivial, can be used to factor n. https://en.wikipedia.org/w/index.php?title=Fermat%27s_factorization_method&oldid=984206978, Articles with unsourced statements from January 2015, Articles with unsourced statements from January 2020, Creative Commons Attribution-ShareAlike License, This page was last edited on 18 October 2020, at 20:39. y β a Third year Department of Information Technology Jadavpur University. A reasonable choice could be By using our site, you Segmented Sieve (Print Primes in a Range), Prime Factorization using Sieve O(log n) for multiple queries, Efficient program to print all prime factors of a given number, Pollard’s Rho Algorithm for Prime Factorization, Convert a String to Character array in Java, Implementing a Linked List in Java using Class, Program to print ASCII Value of a character, Java Program to find largest element in an array, Multiply large integers under large modulo, Dixon's Factorization Method with implementation, Pollard's Rho Algorithm for Prime Factorization, Sum of Factors of a Number using Prime Factorization, Count occurrences of a prime number in the prime factorization of every element from the given range, Trial division Algorithm for Prime Factorization, GCD of two numbers when one of them can be very large, Multiply Large Numbers represented as Strings, Maximum GCD of all subarrays of length at least 2, Print the sequence of size N in which every term is sum of previous K terms, Dijkstra's shortest path algorithm in Java using PriorityQueue, Java Swing | Simple User Registration Form. For example, if n N Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: That difference is algebraically factorable as {\displaystyle X\approx {\sqrt {N/q}}} ) a / < − N Squares are always congruent to 0, 1, 4, 5, 9, 16 modulo 20. and / {\displaystyle a=80} Fermat’s Factorization method for large numbers Last Updated: 17-04-2020 Given a large number N , the task is to divide this number into a product of two factors, using Fermat… {\displaystyle \lfloor ab/\beta \rfloor =t} N = {\displaystyle \gcd(N,cv)=c} = , a square. {\displaystyle a^{2}-n} In the above example, with {\displaystyle Y=\beta y+b} 59 / b s {\displaystyle a+b=101} + 2 If N is prime (so that 2 = − = This is a bad way to prove primality. ) which ends in 4 mod 20 and, if square, b will end in 2 or 8 mod 10. More precisely, if c differs less than , Fermat's method would have found it already. > = steps. a = b {\displaystyle {\left(4N\right)}^{1/4}} {\displaystyle N=2345678917} Fermat's factorization method is also a first step towards understanding c {\displaystyle O(N)} c {\displaystyle a-b=N/(a+b)} , and the factors of 5959 are O − , then + {\displaystyle c} Going up from . 2 The fundamental ideas of Fermat's factorization method are the basis of the quadratic sieve and general number field sieve, the best-known algorithms for factoring large semiprimes, which are the "worst-case". 4 Fermat's method gives factors quickly when they are near the root of the number. b Given a sequence of a-values (start, end, and step) and a modulus, one can proceed thus: But the recursion is stopped when few a-values remain; that is, when (aend-astart)/astep is small. b a Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. ) is known, then the rational number This all suggests a combined factoring method. {\displaystyle Y^{n}/X^{m}\approx q} . from X where β N Fermat's method works best when there is a factor near the square-root of N. If the approximate ratio of two factors ( v / , meaning that {\displaystyle N=\beta z+r} ( / a (A multiple of four is also a difference of squares: let c and d be even.). can be picked near that value. v , d Also, because a's step-size is constant, one can compute successive b2's with additions. . , a Choose some bound = Generally, if the ratio is not known, various 3 The second attempt also fails, because 282 is again not a square. q = {\displaystyle \lfloor N/\beta ^{2}\rfloor -s=xy} 2345678917 m ( must be 1 mod 20, which means that a is 1, 9, 11 or 19 mod 20; it will produce a That procedure first finds the factorization with the least values of a and b. N are squares: It is not necessary to compute all the square-roots of = − a (Unless c divides u or d divides v.) Using the same b N {\displaystyle X\approx N^{m/(m+n)}/q^{1/(m+n)}} {\displaystyle Y/X\approx q}