98 HISTORY OF THE THEORY OF NUMBERS. [CHAP.III Sylvester276 stated that, if pb p2) • • -are the successive primes 2, 3, 5, . . ., where Fk(ri) is a polynomial of degree k with integral coefficients, and the exponent e of the prime p is given by E. Cesaro277 stated Sylvester's269 theorem and remarked that Sn>m— nl is divisible by m— n if m— n is a prime. E. Cesaro278 stated that the prime p divides £m,p_2— 1> £p-i,p+l, and, except when m=p—l, £«,P_I. Also (p. 401), each prime p>(n+l)/2 divides Sp_i,n+l, while a prime p = (n+l)/2 or n/2 divides jSp_i,w+2. 0. H. Mitchell279 discussed the residues modulo k (any integer) of the symmetric functions of 0, 1, . . . , k — 1. To this end he evaluated the residue of (x— a)(x — 0) . . . , where a, /3, . . .are the s-totitives of k (numbers <k 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. Lionnet280 stated and Moret-BIanc proved that, if p = 2n-fl 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+1, n+2, . . . , 2n, are divisible by p. M. d'Ocagne281 proved the first relation of Torelli.271 E. Catalan282 stated and later proved283 that sk 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} skz= — l (mod p). Let p = aatf . . . ; if no one of a—I, 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 is divisible by p; but, if p — 1 divides k+l, $= — ( — 1)* (mod p) . If A; and I are of contrary parity, p divides S. M. d'Ocagne284 proved for Fergola's270 symbol the relation (a. . Jg. . .1. . .v. . .*)-=Z(a. . .f)\g . . .Z)*. ..(*.. .*)', summed for all combinations such that X-fju+ . . . -fp = n. Denoting by a(p) the letter a taken p times, we have (a(p)a&. . J)n= S al'(l(p))'(a&. . J)n~\ t-O »78Nouv. Ann. Math., (2), 6, 1867, 48. *"Ncmv. Corresp. Math., 4, 1878, 401; Nouv. Ann. Math., (3), 2, 1883, 240. 278Nouv. Corresp. Math., 4, 1878, 368. *79Amer. Jour. Math., 4, 1881, 25-38. 280Nouv. Ann. Math., (3), 2, 1883, 384; 3, 1884, 395-6. ™Ibid., (3), 2, 1883, 220-6. Cf. Cesaro, (3), 4, 1885, 67-9. 282Bull. Ac. Sc. Belgique, (3), 7, 1884, 448-9. 28JM6m. Ac. R. Sc. Belgique, 46, 1886, No. 1, 16 pp. 2MNouv. Ann. Math., (3), 5, 1886, 257-272.