24 HISTORY OF THE THEORY OF NUMBERS. [CHAP, i H. LeLasseur found after131 1878 and apparently just before182 1882 that 2n— 1 has the prime factor 11447 if n = 97, 15193 if n = 211, 18121 if n = 151, 18287 if n = 223, and that there is no divisor < 30000 of 2n-l for the 24 prime values of n, rig 257, which remain in doubt, viz. [cf. Lucas136], 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. Carvallo133 attempted again114 to prove the non-existence of odd perfect numbers ynzv . . . ur, where y, . . . u are distinct odd primes. He began by noting that one and only one of the exponents n, . . . , r is odd [Euler98]. Let y<z< . . .<u, and call their number fjL. From the definition of a perfect number, y-i/yn tt-W^o y u 2/-i •;• w-i ' 2/-r"w-i The fractions in this inequality form a decreasing series. Hence Thus u(2— k)<2. By a petitio principii (the division by 2 — fc, not known to be positive), it was concluded (p. 10) that 2 [This error, repeated on p. 15, was noted by P. Mansion.134] For a given n, there is at most one prune between the two limits (of difference <2) for y. A superior limit is found for z as a function of y. An incomplete computation is made to show that, if ju> 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, 229 — 1 and 241 — 1 are entered as primes [contrary to Euler83 and Plana110]. E. Catalan135 stated that 2P — 1 is a prime if p is a prime of the form 2X-1. If correct this would imply thaf2127-l is a prime [cf. Catalan116]. E. Lucas136 repeated the remark of LeLasseur132 on the 24 prime values of n^257 for which the composition of 2n-l is in doubt. According to a 13lSince these four values of n are included in the list by Lucasm of the 28 values of n^257 for which the composition of 2n-l is unknown. Cf. Lucas123, p. 236. "'Lucas, Recreations math., 1, 1882, 241; 2, 1883, 230. Later, Lucas126 credited LeLasseur with these four cases as well as n = 73 [Euler83! and n = 79, 113, 233 [cf. Reuschle101]. The last four cases were given by Lucas124, while the last three do not occur in the table (Lucas124, pp. 788-9) by LeLasseur of the proper divisors of 2f>—l for each odd n, n<79, and for a few larger composite n's. The last three were given also by Lucas113 (p. 236) without reference. U3Th4orie des nombres parfaits, par M. Jules Carvallo, Paris, 1883, 32 pp. "'Mathesis, 6, 1886, 147. "'Melanges Math., Bruxelles, 1, 1885, 376. 1MMathesis, 6, 1886, 146.