Factorization of the eleventh Fermat number

113. R. P. Brent, Factorization of the eleventh Fermat number (preliminary report), AMS Abstracts 10 (1989), 89T-11-73.

Abstract: dvi (3K), pdf (32K), ps (28K).

Paper: dvi (2K), pdf (19K), ps (14K).


Of the Fermat numbers Fn =  22n + 1, only F1 to F4 are known to be prime; certainly F5 to F21 are composite. However, the only complete factorizations known until now are those of

This abstract announces the complete factorization of the 617-digit Fermat number F11 = 2211 + 1:

F11 = 319489 . 974849 .  167988556341760475137 . 3560841906445833920513 .  p564

where the two 6-digit factors were already known (Cunningham, 1899), the 21-digit and 22-digit prime factors were found using the two-phase elliptic curve algorithm on a Fujitsu VP100 computer, and p564 is a 564-decimal digit prime (primality proof by F. Morain, using the method of Atkin).


For a description of the computational method and related work, see [97, 115, 120, 161].

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