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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