26 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. I
Cl. Servais145 republished the proofs by Novarese143 and proved that ambn is not perfect if a and 6 are odd prunes. For, by the equations [Nocco111]
o"+1-l = bn(a-we obtain, by subtraction,
Thus 2am> bn. Since a^ 3, am+1^ 3am> am+bn> a+b -1. He next proved that, if an odd perfect number is divisible by only three distinct primes a, b, c, two of them are 3 and 5, since [as by Carvallo133]
Taking a = 3, 6 = 5, we have c<16, whence c = 7, 11, or 13. He quoted from a letter from Catalan that the sum of the reciprocals of the divisors of a perfect number equals 2.
E. Cesaro146 proved that in an odd perfect number containing n distinct prime factors, the least prime factor is ^n^/2.
Cl. Servais147 showed that it does not exceed n since, if a<6<c< ...,
6 a+1 c a+2
b — 1 a c—1 a+11'
a b a a+1 a+2 a+n-1
a—l'b — l'" a— 1" a "a+1 ' * ' a+n— 2
whence 2(a— l)<a+n— 1, a<n+l. If Z is the (m— l)th prime factor and 5 is the Twth, and if
r 8 8+1 s+n—m L(n—m)+2
«-!' s '"s+n-m+r ' 2-L
J. J. Sylvester148 reproduced Euler's97 proof that every even perfect number is of Euclid's type. From the fact that -|4<2, he concluded that there is no odd perfect number ambn. For the case of three prime factors he obtained the result of Servais145 in the same manner. He proved that no odd perfect number is divisible by 105 and stated that there is none with fewer than six distinct prime factors.
Sylvester149 and Servais150 gave complete proofs that there exists no odd perfect number with only three distract prime factors.
""Mathesis, 7, 1887, 228-230. ™Ibid., 245-6. "'Mathesis, 8, 1888, 92-3.
"•Nature, 37, Dec. 15, 1887, 152 (minor correction, p. 179); Coll. Math. Papers, 4, 1912, 588. M9Comptes Rendus Paris, 106, 1888, 403-5 (correction, p. 641); reproduced with notes by P. Mansion, Mathesis, 8, 1888, 57-61. Sylvester's Coll. Math. Papers, 4, 1912, 604, 615. ""Mathesis, 8, 1888, 135.