Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 27 
Sylvester ^^^ proved there is no odd perfect number not divisible by 3 
with fewer than eight distinct prime factors. 
Sylvester^^^ proved there is no odd perfect number with four distinct 
prime factors. 
Sylvester^^^ 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 investi- 
gation of this question: For r an integer other than 1 or —1, the sum 
l+r+r^+ . . . -\-r^~^ 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. Catalan ^^^ 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 
lTl3-- I ^2 ^^^^"3 5 7 11 • I <2 3 5 7<^-^^^^- 
By Legendre's table IX, Theorie des nombres, ed. 2, 1808; ed. 3, 1830, of 
the values of P{w) up to w; = 1229, we see that I ^ 127. But 127 is the 27th 
prime >7. 
R. W. D. Christie^^^ erroneously considered 2^^ — 1 and 2^^ — 1 as primes. 
E. Lucas^^® 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 7k='= 1 ; 
any one except 6 has the remainder 1,2, 3, or 8 when divided by 13, etc. 
E. Lucas^" reproduced his^^^ proofs and the proof by Euler,^^ and gave 
(p. 375) a list of known factorizations of 2" — 1. 
Genaille^^^ stated that his machine "piano arithm^tique " gives a prac- 
tical means of applying in a few hours the test by Lucas {ibid., 5, 1876, 61) 
for the primality of 2" — 1 . 
J. Fitz-Patrick and G. ChevreP^^ stated that 2^8(229-1) is perfect. 
E. Fauquembergue^®^ found that 2®'' — 1 is composite by a process not 
yielding its factors [cf. Mersenne,^" Lucas,^^^ Cole^'^^]. 
A. Cunningham^^^ called 2^ — 1 a Lucassian if p is a prime of the form 
4A;+3 such that also 2p+l is a prime, stating that Lucas^^^ had proved that 
2'' — 1 has the factor 2p+l. Cunningham listed all such primes p<2500 
i"Comptes Rendus Paris, 106, 1888, 448-450; Coll. M. Papers, IV, 609-610. 
^mid., 522-6; Coll. M. Papers, IV, 611-4. 
""Nature, 37, 1888, 417-8; Coll. M. Papers, IV, 625-9. 
""Mathesis, 8, 1888, 112-3. M6m. soc. sc. Ukge, (2), 15, 1888, 205-7 (Melanges math., III). 
»*Math. Quest. Educat. Times, 48, 1888, p. xxxvi, 183; 49, p. 85. 
"«Mathe8is, 10, 1890, 74-76. 
"'Theorie des nombres, 1891, 424-5. 
"'Assoc, frang. avanc. sc, 20, I, 1891, 159. 
"'Exercices d'Arith., Paris, 1893, 363. 
""L'interm^diaire des math., 1, 1894, 148; 1915, 105, for representations by u*+67t;*. 
""British Assoc. Reports, 1894, 563. 
