CHAP. I] PEEFECT, MULTIPLY PERFECT, AND AMICABLE NUMBEBS. 27 Sylvester151 proved there is no odd perfect number not divisible by 3 with fewer than eight distinct prime factors. Sylvester152 proved there is no odd perfect number with four distinct prune factors. Sylvester153 spoke of the question of the non-existence of odd perfect numbers as a "problem of the ages comparable in difficulty to that which previously to the labors of Hermite and Lindemann environed the subject of the quadrature of the circle." He gave a theorem useful, for the investigation of this question: For r an integer other than 1 or — 1, the sum l+r+r2-f-.. .+rp~*1 contains at least as many distinct prime factors as p contains divisors >1, with a possible reduction by one in the number of prime factors when r = —2, p even, and when r=2, p divisible by 6. E. Catalan154 proved that if an odd perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime factors and thus has at least 45 digits. In fact, the usual inequality gives El? Izl^l, p,n_2A6.10 !-!<! ? 4 8 1113-" I <2) P(Z)"~3 5 7 ir"~T<2*3*5 7<a2285' By Legendre's table IX, ThSorie des nombres, ed. 2, 1808; ed. 3, 1830, of the values of P(w) up to w = 1229, we see that Z ^ 127. But 127 is the 27th prune >7. R. W. D. Christie155 erroneously considered 241 —1 and 247 — 1 as primes. E. Lucas156 proved that every even perfect number, aside from 6 and 496, ends with 16, 28, 36, 56, or 76; any one except 28 is of the form 7/c± 1; any one except 6 has the remainder 1,2, 3, or 8 when divided by 13, etc. E. Lucas157 reproduced his156 proofs and the proof by Euler,98 and gave (p. 375) a list of known factorizations of 2n —1. Genaille158 stated that his machine "piano ari thine* tique" gives a practical means of applying in a few hours the test by Lucas (ibid., 5, 1876, 61) for the primality of 2n— 1. J. Fitz-Patrick and G. Chevrel159 stated that 228(229-l) is perfect. E. Fauquembergue160 found that 267 —1 is composite by a process not yielding its factors [cf. Mersenne,60 Lucas,115 Cole173]. A. Cunningham161 called 2P —1 a Lucassian if p is a prime of the form 4fc+3 such that also 2p+1 is a prime, stating that Lucas123 had proved that 2P —1 has the factor 2p-fl. Cunningham listed all such primes p<2500 1B1Comptes Rendus Paris, 106, 1888, 448-450; Coll. M. Papers, IV, 609-610. lM/6td., 522-6; Coll. M. Papers, IV, 611-4. ""Nature, 37, 1888, 417-8; Coll. M. Papers, IV, 625-9. 1MMathesis, 8, 1888, 112-3. M6m. soc. sc. Liege, (2), 15, 1888, 205-7 (Melanges math., III). 1MMath. Quest. Educat. Times, 48, 1888, p. xxxvi, 183; 49, p. 85. "•Mathesis, 10, 1890, 74-76. "'Theorie des nombres, 1891, 424-5. "8Assoc. franc, avanc. sc., 20, I, 1891, 159. "•Exercices d'Arith., Paris, 1893, 363. 160L'interme*diaire des math., 1, 1894, 148; 1915, 105, for representations by us+67t>s. lfllBritish Aseoc. Reports, 1894, 563.