113 – Appeared in AMS Abstracts 10(1989), 89T-11-73

RICHARD P. BRENT, Australian National University, Canberra, ACT 2601, Australia. \underline{Factorization of the Eleventh Fermat number}. Preliminary report.

Of the Fermat numbers F_{n} = 2^{2^{n}} + 1, only F_{1} to F_{4} are known to be prime; certainly F_{5} to F_{21} are composite. However, the only complete factorizations known until now are those of F_{5} (Euler), F_{6} (Landry), F_{7} (Morrison and Brillhart, 1975), and F_{8} (Brent and Pollard, 1981). This abstract announces the complete factorization of the 617-digit Fermat number F_{11} = 2^{2^{11}} + 1. In fact

F_{11} = 319489.974849.167988556341760475137.3560841906445833920513.p_{564}

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 p_{564} is a 564-decimal digit prime (primality proof by F. Morain, using the method of Atkin). The provision of computer time by the ANU Supercomputer Facility is gratefully acknowledged.

[Submitted to AMS Abstracts 21 December 1988.]