CHAP.I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 31 A. Cunningham190 found the factor 730753 of 2173-1. V. Ramesam191 verified that the quotient of 271-1 by the factor 228479 [Cunningham181] is the product of the primes 48544121 and 212885833. A. Aubry192 stated erroneously that "Mersenne affirmed that 2n—1 is a prime, for 7^257, only for n = l, 2, 3, 4, 8, 10, 12, 29, 61, 67, 127, 257 (which has now been almost proved); this proposition seems to be due to Frenicle.57" What Mersenne60 actually stated was that the first 8 perfect numbers occur at the lines marked 1, 2, 3, 4, 8, etc., in the table by Bungus. A. Cunningham192* noted that M113, MM, M25i have the further factors 23279-65993, 55871, 54217, respectively. Cf. Reuschle108, Lucas123. A. G£rardin192& noted that there is no divisor < 1000000 of the composite Mersenne numbers not already factored. Let d denote the least divisor of 2«-l, q a prime ^257. If g = 60w+43, then fe47 (mod 96), except for the cases given by Euler's83 theorem (verified for 43, 163, 223). If g=40w+33, fe7 (mod 24), verified for 73,113, 233. If g = 30w+l, d=l (mod 24), verified for 31, 61, 151, 181, 211. E. Fauquembergue192c proved that 2101~1 is composite by means of Lucas' test with 4, 14, 194,..., written to base 2 (Ch. XVII). L. E. Dickson193 called a non-deficient number primitive if it is not a multiple of a smaller non-deficient number, and proved that there is only a finite number of primitive non-deficient numbers having a given number of distinct odd prime factors and a given number of factors 2. As a corollary, there is not an infinitude of odd perfect numbers with any given number of distinct prune factors. There is no odd abundant number with fewer than three distinct prime factors; the primitive ones with three are 335-7, 32527, 325-72, 335211, 355213, 345313, 3452132, 335313l There is given a list of the numerous primitive odd abundant numbers with four distinct prime factors and lists of even non-deficient numbers of certain types. In particular, all primitive non-deficient numbers < 15000 are determined (23 odd and 78 even). In view of these lists, there is no odd perfect number with four or fewer distinct prime factors (cf. Sylvester148"153). A. Cunningham194 gave a summary of the known results on the composition of the 56 Mersenne numbers Mq — 2q—1, q a prime ^257. Of these, 12 have been proved prime: Mv q = l, 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 127; while 29 of them have been proved composite. Thus only 15 remain in 190British Assoc. Reports, 1912, 406-7. Sphinx-Oedipe, 7, 1912, 38 (1910, 170, that 730753 is a possible factor). Cf. Cunningham104. 181Nature, 89, 1912, p. 87; Sphinx-Oedipe, 1912, 38. Jour, of Indian Math. Soc., Madras, 4 1912, 56. 1MOeuvres de Fermat, 4, 1912, 250, note to p. 67. 1920 Mem. and Proc. Manchester Lit. and Phil. Soc., 56, 1911-2, No. 1. 1826 Sphinx-Oedipe, 7, 1912, nume'ro special, 15-16. M'lbid., Nov., 1913, 176. 183Amer. Jour. Math., 35, 1913, 413-26. 194Proc. Fifth International Congress, I, Cambridge, 1913, 384-6. Proc. London Math. Soc., (2), 11, 1913, Record of Meeting, Apr. 11, 1912, xxiv. British Assoc. Reports, 1911, 321. Math. Quest. Educat. Times, (2), 23, 1913, 76.