434[F].—D. H. LEHMER,
"On the factors of
2^{n} ± 1
," Amer. Math. So., Bull., v. 53, Feb. 1947,
p. 164167,
15.1 × 24 cm.
Professor Lehmer gives factors of
2^{n}  1,
in 32 cases, for values of n from 113 to 489, and of
2^{n} + 1
in 44 cases, for values of n from 91 to 500. This list for
n < 500 was intended to supplement the
fundamental table of CUNNINGHAM &
WOODALL^{1}
and the addenda to this list found by
KRAÏTCHIK^{2}.
It is believed that all factors under
10^{6}
have now been found, and that any other factors of
2^{n}  1
for n < 300, or of
2^{n} + 1
for n < 150, lie beyond 4538800.
Eight complete factorizations, n varying
from 91 to 170, are given; the fifth of these for
2^{123} + 1
has been already noted in MTE 107. The first and eigth correct errors
in Kraïtchik and in Cunningham & Woodall.
Eleven of the new factors given by Lehmer pertain
to Mersenne numbers
2^{p}  1,
p a prime not greater than 257. These factors are included in
the range p = 113 (now completely factored) to
p = 223. Of the 55 Mersenne numbers 12 are prime
(p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127), 14 are
composite and completely factored, for 9 two or more prime factors
are known, for 8 only one prime factor is known, 11 are composite but
no factor known, and in one case (p = 193) the
character is unknown. As indicated above, any other factor now
discovered for a Mersenne number must be greater than 4538800.
Professor H. S. UHLER completed
the proof that M_{199}
was composite on July 27, 1946 (Amer. Math. Soc. Bull.,
v. 53, 1947, p. 163164); and that
M_{227} was composite
on June 4, 1947; see also MTAC, v. 1,
p. 333
(M_{157}),
404
(M_{167}), v. 2,
p. 94
(M_{229}).
In the article here reviewed D. H. L. checked the last two
results at which Uhler had arrived, by showing that
M_{167} had
the factor 2349023 and
M_{229} the factor
1504073.
^{1} A. J. C.
CUNNINGHAM & H. J. WOODALL,
Factorisation of
(y^{n} ± 1),
London, 1925.
^{2} M.
KRAÏTCHIK, (a) Recherches sur la
Théorie des Nombres, v. 2, Paris, 1929; (b)
"Factorisation de
2^{n} ± 1,"
Sphinx, v. 8, 1938, p. 148150.
