24 History of the Theory of Numbers. [Chap, i 
H. LeLasseur found after^^^ 1878 and apparently just before^^^ 1882 
that 2'*-l has the prime factor 11447 if n = 97, 15193 if n = 211, 18121 if 
71 = 151, 18287 if n = 223, and that there is no divisor <30000 of 2"-l for 
the 24 prime values of n, n^257, which remain in doubt, viz. [cf. Lucas^^^], 
61, 67, 71, 89, 101, 103, 107, 109, 127, 137, 139, 149, 
157, 163, 167, 173, 181, 193, 197, 199, 227, 229, 241, 257. 
J. Carvallo^^^ attempted again^^* to prove the non-existence of odd 
perfect numbers y^'z^ . . .u\ where y,. . .u are distinct odd primes. He began 
by noting that one and only one of the exponents n, . . ., r is odd [Euler^^]. 
Let y<z< . . .<u, and call their number jjl. From the definition of a 
perfect number, 
y — l "' u — 1 ' y — l'u — l 
The fractions in this inequality form a decreasing series. Hence 
fe)'>^. ^<A' -~i>^' H^r 
Thus w(2 — A;)<2. By a petitio principii (the division by 2— A:, not known 
to be positive), it was concluded (p, 10) that 
^<2i:^' ^^<2, y> 2i/(M-i)_i - 
[This error, repeated on p. 15, was noted by P. Mansion. ^^] For a 
given n, there is at most one prime between the two limits (of difference < 2) 
for y. A superior limit is found for 2 as a function of y. An incomplete 
computation is made to show that, if /x>8, z <y-\-l. 
It is shown (p. 7) that an odd perfect number has a prime factor greater 
than the prime factor w entering to an odd power, since w+l divides the 
sum of the divisors. In a table (p. 30) of the first ten perfect numbers, 
2^^ — 1 and 2*^ — 1 are entered as primes [contrary to Euler^^ and Plana^^°]. 
E. Catalan^^^ stated that 2^ — 1 is a prime if p is a prime of the form 
2^ — 1. If correct this would imply that 2^^^ — 1 is a prime [cf. Catalan^^^]. 
E. Lucas^^^ repeated the remark of LeLasseur^^^ on the 24 prime values 
of n^257 for which the composition of 2^ — 1 is in doubt. According to a 
"iSince these four values of n are included in the list by Lucas^** of the 28 values of n ^ 257 for 
which the composition of 2"—! is unknown. Cf. Lucas^^^ p. 236. 
"2Lucas, Recreations math., 1, 1882, 241; 2, 1883, 230. Later, Lucas^^s credited LeLasseur 
with these four cases as well as n = 73 [Eulers^] and n = 79, 113, 233 [cf. Reuschlei"]. 
The last four cases were given by Lucas"*, while the last three do not occur in the table 
(Lucasi24^ pp 7gg_9) by LeLasseur of the proper divisors of 2"— 1 for each odd n, n<79, 
and for a few larger composite n's. The last three were given also by Lucas"^ (p. 236) 
without reference. 
'"Th^orie des nombres parfaits, par M. Jules Carvallo, Paris, 1883, 32 pp. 
"<Mathesis, 6, 1886, 147. 
'"Melanges Math., Bruxelles, 1, 1885, 376. 
'"Mathesis, 6, 1886, 146. 
