Factorization of the tenth Fermat number

161. R. P. Brent, Factorization of the tenth Fermat number, Mathematics of Computation 68 (1999), 429-451. MR 99e:11154.

Preliminary (slightly different) version available as Factorization of the tenth and eleventh Fermat numbers, Technical Report TR-CS-96-02, CSL, ANU, Feb. 1996, 25 pp.

Abstract: dvi (3K), pdf (82K).

Paper: dvi (53K), pdf (363K), ps (116K).

Technical Report: dvi (60K), pdf (539K), ps (190K).

Transparencies: dvi (12K), pdf (218K), ps (56K).

Abstract for Paper

We describe the complete factorization of the tenth Fermat number F10 by the elliptic curve method (ECM). The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the factorization of other Fermat numbers by ECM, and summarize the complete factorizations of F5 , ... , F11.

Abstract for Technical Report

We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27-digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391-decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40-digit factor of the tenth Fermat number was found after about 140 Mflop-years of computation. We discuss aspects of the practical implementation of ECM, including the use of special-purpose hardware, and note several other large factors found recently by ECM.

Comments

A slight variation of the Fortran program used to factor F10 is available here.

Related papers include:

Go to next publication

Return to Richard Brent's index page