Factorization of the tenth Fermat number

Abstract

Abstract. We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 , . . . , F 11 .

Three new factors of Fermat numbers

“…Various practical refinements were suggested by Brent [1], Montgomery [24,25], and Suyama [32]. We refer to [3,14,22,26,31] for a description of ECM and some of its implementations.…”

“…Brent [2,3,4] completed the factorization of F 10 (by finding a 40-digit factor) and F 11 . He also "rediscovered" the 49-digit factor of F 9 and the five known prime factors of F 12 .…”

Backtracking Based Integer Factorisation, Primality Testing and Square Root Calculation

“…This method initially was proposed by Pollard [8]. Some variations, improvements and optimisations of Rho methods are proposed in - [9], [2], [1] etc.…”

Finding special factors of values of polynomials at integer points

“…For example F 7 was factored in 1970 with the continued fraction method, the first known factoring algorithm of subexponential complexity [2]. The fastest currently known algorithms, the elliptic curve method and (special) number field sieve, were used to factor F 10 and F 9 respectively (see [1] and [7]). Unfortunately, since 1999 no more Fermat numbers were completely factored, and because of their rapid growth it seems that some dramatic development of factoring methods (or a lot of luck) is needed to factorize F 12 , the smallest Fermat number which complete factorization is unknown.…”

“…We conducted a search of the solutions (a 1 , a 2 ) of the equation (14) with c 2 31 and a 1 2 1000 . This gave us 2698 pairs (a 1 , a 2 ) ∈ F P .…”