98 History of the Theory of Numbers. [Chap, hi 
Sylvester"^ stated that, if pi, p2, . . .are the successive primes 2, 3, 5, . . ., 
^'^^ ^T^^A^ ^'-'^''^' 
where Fk{n) is a polynomial of degree k with integral coefficients, and the 
exponent e of the prime p is given by 
E. Ces^ro^" stated Sylvester's^®^ theorem and remarked that <S„.„— n! 
is divisible by w —n if m—n is a prime. 
E. Ces^ro"^ stated that the prime p divides /S^.p-2 — 1, *Sp_i.p+l, and, 
except when m = p — l, S^.p-i- Also (p. 401), each prime p>{n+l)/2 
divides *Sp_i.„H-l, while a prime p = (n+l)/2 or n/2 divides »Sp_i.„+2. 
O. H. Mitchell"^ discussed the residues modulo k (any integer) of the 
symmetric functions of 0, 1, . . . , /c — 1. To this end he evaluated the residue 
of (x— a)(x— /3) . . ., where a, /3, . . .are the s-totitives of k (numbers<A: which 
contain s but no prime factor of k not found in s) . The results are extended 
to the case of moduli p, f{x), where p is a prime [see Ch. VIII]. 
F. J. E. Lionnet^^° stated and Moret-Blanc proved that, if p = 2n+l is 
a prime> 3, the sum of the powers with exponent 2a (between zero and 2n) 
of 1, 2, . . . , n, and the like sum for n-\-l, n+2, . . . , 2n, are divisible by p. 
M. d'Ocagne^^^ proved the first relation of Torelli.^^^ 
E. Catalan^^^ stated and later proved^^^ that s^ is divisible by the prime 
p>k-\-l. If p is an odd prime and p — 1 does not divide k, Sk is divisible 
by p; while if p — 1 divides k, Sk= — l (mod p). Let p = a''b^ . . . ; if no one 
of a — 1, 6 — 1,. . . divides k, Sk is divisible by p; in the contrary case, not 
divisible. If p is a prime >2, and p — 1 is not a divisor of k-\-l, then 
^ = l^(p-l)'+2*(p-2)^+ . . .+{p-iyv 
is divisible by p; but, if p — 1 divides k+l, S= — { — iy (mod p). If k and I 
are of contrary parity, p divides S. 
M. d'Ocagne^^ proved for Fergola's^'^" symbol the relation 
(a. . .fg. . .1. . .V. . .zr^Xia. . .frig. . .^)^ ..{v. . .z)", 
summed for all combinations such that X+/i+. . .-\-p = n. Denoting by 
a^^^ the letter a taken p times, we have 
i=0 
"«Nouv. Ann. Math., (2), 6, 1867, 48. 
»"Nouv. Correap. Math., 4, 1878, 401; Nouv. Ann. Math., (3), 2, 1883, 240. 
"8Nouv. Coiresp. Math., 4, 1878, 368. 
"»Amer. Jour. Math., 4, 1881, 25-38. 
"ONouv. Ann. Math., (3), 2, 1883, 384; 3, 1884, 395-6. 
"»/6id., (3), 2, 1883, 220-6. Cf. Ces^ro, (3), 4, 1885, 67-9. 
2«BuU. Ac. Sc. Belgique, (3), 7, 1884, 448-9. 
"'M6m. Ac. R. Sc. Belgique, 46, 1886, No. 1, 16 pp. 
»**Nouv. Ann. Math., (3), 5, 1886, 257-272. 
