## 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
*M*_{k} =
2^{k} - 1.
They were found by the Pollard rho,
"*p* - 1", and
"*p* + 1"
methods on a Univac 1100/42.
The factorizations of *M*_{k}
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 *M*_{337} the largest factor should be "p68",
not "prp67" (fixed in the online version).
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