Factorization of the Tenth and Eleventh Fermat Numbers

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-decimal 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. CommentsOnly the Abstract is given here. The full report appeared as [1]. A shorter version (omittingthe factor of F13) appeared as [2]. The factor of F13 is mentioned in [3]. References[1] R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Technical Report TR-CS-96-02, CSL,ANU, Feb. 1996, 25 pp. ftp://discus.anu.edu.au/pub/Brent/rpb161tr.dvi.gz. rpb161tr[2] R. P. Brent, “Factorization of the tenth Fermat number”, Math. Comp. 68 (1999), 429–451. rpb161[3] R. P. Brent, R. E. Crandall, K. Dilcher and C. Van Halewyn, “Three new factors of Fermat numbers”,Mathematics of Computation S 0025-5718(00)01207-2 (published electronically 1 March 2000). rpb175 Computer Sciences Laboratory, Australian National University, Canberra, ACT 0200E-mail address: [email protected] Date: 21 February 1996.1991 Mathematics Subject Classification. 11Y05, 11B83, 11Y55; Secondary 11-04, 11A51, 11Y11, 11Y16,14H52, 65Y10, 68Q25.