New factors of Mersenne numbers
67. R. P. Brent,
New factors of Mersenne numbers (preliminary report),
AMS Abstracts 2 (1981), 367, 81T-10-246;
part II, ibid 3 (1982), 132, 82T-10-22;
part III, ibid 4 (1983), 197, 83T-10-138.
Paper as submitted (parts I-III):
pdf (131K).
Abstract
These brief reports list 35 new factors (not counting the largest factors)
of Mersenne numbers
Mk =
2k - 1.
They were found by the Pollard rho,
"p - 1", and
"p + 1"
methods on a Univac 1100/42.
The factorizations of Mk
for k = 227, 229, 283, 331, 337, 373, 379,
409, 431, 439, 443, and 577 are complete.
Comments
In three cases (k = 373, 379, 577)
the largest factors were proved prime by
H. Cohen and H. W. Lenstra;
in four other cases (k = 283, 331, 337, 409)
the largest factors were proved prime
by H. C. Williams and the author.
In two cases (k = 617, 797) the new factors were
found independently by H. Suyama and the author.
The factors are all included in the
Cunningham tables; see also my
factor table page.
At least two of the factors
(p21 for k = 439 and p20 for k = 509)
were found by the author's implementation of the
"p + 1" method,
which is analogous to Pollard's "p - 1" method.
See
Zimmermann's list
for other factors found by the "p + 1" method.
Erratum
For M337 the largest factor should be "p68",
not "prp67" (fixed in the online version).
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