434[F].—D. H. LEHMER,
"On the factors of
2
n
± 1
," Amer. Math. So., Bull., v. 53, Feb. 1947,
p. 164-167,
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. 163-164); 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. 148-150.
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