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).
Abstract
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
- F5 (Euler),
- F6 (Landry),
- F7 (Morrison and Brillhart, 1975), and
- F8
(Brent and Pollard, 1981).
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).
Comments
For a description of the computational method and related work,
see
[97,
115,
120,
161].
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