26 History of the Theory of Numbers. [Chap, i 
CI. Servais"^ republished the proofs by Novarese^"*^ and proved that 
a'"6'* is not perfect if a and h are odd primes. For, by the equations [Nocco"^] 
a-+i_l = 6"(a-l), b''+^-l=2a"'(6-l), 
we obtain, by subtraction, 
Thus2a'">6". Since a^3, a'"+^^3a"'>a'"+6'*>a+6-l. He next proved 
that, if an odd perfect number is divisible by only three distinct primes a, 
h, c, two of them are 3 and 5, since [as by Carvallo^^^] 
04)04)04)<l 
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. Cesaro^*^ proved that in an odd perfect jiumber containing n distinct 
prime factors, the least prime factor is ^n\/2. 
CI. Servais^^^ showed that it does not exceed n since, if a<h<c< . . . , 
b a+1 c a-\-2 
6 — 1 a c — \ a+1 
ah a a+1 a+2 a-\-n — \ 
2i^. . . ^ . . . . > 
a—lb—1 ' a— r a a+1 a+n— 2 
whence 2(a — l)<a+n — 1, a<n+l. If I is the (m — l)th prime factor and 
s is the 772th, and if 
a b 
a-l'b-l"l-l 
then 
^L<2, 
s+1 s+n—m _. ^L{n—m)-\-2 
>2, s< 
s — l s ' ' ' s-\-n—m-\-l ' 2—L 
J. J. Sylvester ^'^^ reproduced Euler's^^ proof that every even perfect 
number is of EucUd's type. From the fact that ■|.|-<2, he concluded that 
there is no odd perfect number a'"6'*. For the case of three prime factors 
he obtained the result of Servais^'*^ 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. 
Sylvester^^^ and Servais^^*^ gave complete proofs that there exists no odd 
perfect number with only three distinct prime factors. 
i«Mathesis, 7, 1887, 228-230. 
»«/6id., 245-6. 
"'Mathesis, 8, 1888, 92-3. 
"8Nature, 37, Dec. 15, 1887, 152 (minor correction, p. 179); Coll. Math. Papers, 4, 1912, 588. 
"'Comptes 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. 
