434[F].—D. H. LEHMER,
"On the factors of
2n ± 1
," Amer. Math. So., Bull., v. 53, Feb. 1947,
p. 164-167,
15.1 × 24 cm.
Professor Lehmer gives factors of
2n - 1,
in 32 cases, for values of n from 113 to 489, and of
2n + 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 &
WOODALL1
and the addenda to this list found by
KRAÏTCHIK2.
It is believed that all factors under
106
have now been found, and that any other factors of
2n - 1
for n < 300, or of
2n + 1
for n < 150, lie beyond 4538800.
Eight complete factorizations, n varying
from 91 to 170, are given; the fifth of these for
2123 + 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
2p - 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 M199
was composite on July 27, 1946 (Amer. Math. Soc. Bull.,
v. 53, 1947, p. 163-164); and that
M227 was composite
on June 4, 1947; see also MTAC, v. 1,
p. 333
(M157),
404
(M167), v. 2,
p. 94
(M229).
In the article here reviewed D. H. L. checked the last two
results at which Uhler had arrived, by showing that
M167 had
the factor 2349023 and
M229 the factor
1504073.
1 A. J. C.
CUNNINGHAM & H. J. WOODALL,
Factorisation of
(yn ± 1),
London, 1925.
2 M.
KRAÏTCHIK, (a) Recherches sur la
Théorie des Nombres, v. 2, Paris, 1929; (b)
"Factorisation de
2n ± 1,"
Sphinx, v. 8, 1938, p. 148-150.
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