Chap.i] Perfect, Multiply Perfect, and Amicable Numbers. 31 
A. Cunningham^^o found the factor 730753 of 2^^^-l. 
V. Ramesam^^i verified that the quotient of 2"^-l by the factor 228479 
[Cunningham^^^] is the product of the primes 48544121 and 212885833. 
A. Aubry^^^ stated erroneously that ''Mersenne affirmed that 2" — 1 is 
a prime, for n^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.^'" What Mersenne^" 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. Cunningham^^^'' noted that M113, M151, M251 have the further factors 
23279-65993, 55871, 54217, respectively. Cf. Reuschle^o^ Lucas^^s 
A. Gerardin^^-^ noted that there is no divisor < 1000000 of the composite 
Mersenne numbers not already factored. Let d denote the least divisor 
of 2«- 1, g a prime ^257. li q = 60z^+43, then d=47 (mod 96), except for 
the cases given by Euler's^^ theorem (verified for 43, 163, 223). If 
5 = 40w+33, d=7 (mod 24), verified for 73, 113, 233. If 5 = 30m+l, d=l 
(mod 24), verified for 31, 61, 151, 181, 211. 
E. Fauquembergue^^^'' proved that 2^°^ — 1 is composite by means of 
Lucas' test with 4, 14, 194,. . ., written to base 2 (Ch. XVII). 
L. E. Dickson^^^ 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 prime factors. There is no odd abundant number with 
fewer than three distinct prime factors; the primitive ones with three are 
3^5-7, 32527, 325.72, 3^5211, 3^13, 3*5^13, 3*52132, 3^5^132. 
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 . Sylvester^*^"^^^) . 
A. Cunningham^^* gave a summary of the known results on the composi- 
tion of the 56 Mersenne numbers Mq = 2^ — 1, q a prime ^257. Of these, 
12 have been proved prime: M^, 5 = 1, 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 127; 
while 29 of them have been proved composite. Thus only 15 remain in 
""British Assoc. Reports, 1912, 406-7. Sphinx-Oedipe, 7, 1912, 38 (1910, 170, that 730753 
is a possible factor). Cf. Cunningham"*. 
i"Nature, 89, 1912, p. 87; Sphinx-Oedipe, 1912, 38. Jour, of Indian Math. Soc, Madras, 4 
1912, 56. 
"K)euvres de Fermat, 4, 1912, 250, note to p. 67. 
"2" Mem. and Proc. Manchester Lit. and Phil. Soc, 56, 1911-2, No. 1. 
i«* Sphinx-Oedipe, 7, 1912, num^ro special, 15-16. 
"'•^ Ibid., Nov., 1913, 176. 
i«Amer. Jour. Math., 35, 1913, 413-26. 
"*Proc. 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. 
