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.
 R. C. A.
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.