96 HlSTOKY OF THE THEOBY OF NUMBEES. [CHAP. Ill J. A. Serret256 concluded by applying Newton's identities to (x—1)... (x—p+l)=0 that sn=0 (mod p) unless n is divisible by p—1. J. Wolstenholme267 proved that the numerators of are divisible by p2 and p respectively, if p is a prune > 3. Proofs have also been given by C. Leudesdorf258, A, Rieke,259 E. Allardice,260 G. Osborn,261 L. Birkenmajer,262 P. Niewenglowski,263 N. Nielsen,264 H. Valentiner,285 and others.266 V. A. Lebesgue267 proved that sm is divisible by p if m is not divisible by p— 1 by use of the identities (n+1) £ *(*+!) . . .(*+n-l) = »(*+!) . . .(s+n) (n-1,. . ., p-1). jb-i P. Frost268 proved that, if p is a prime not dividing 22r— 1, the numerators of or2r, 02,— i, p(2r— I)<72r+2<r2r_i are divisible by p, p2, p3, respectively, where The numerator of the sum of the first half of the terms of <r2r is divisible by p; likewise that of the sum of the odd terms. J. J. Sylvester269 stated that the sum SB, m of all products of n distinct numbers chosen from 1,. . ., m is the coefficient of tn in the expansion of (l+t)(l+2t) . . . (1+mt) and is divisible by each prime >n+l contained in any term of the set w— ?i-f 1, . . . , m, m+l. E. Fergola270 stated that, if (a, 6,..., l)n represents the expression obtained from the expansion of (a-\-b+ . . . -H)n by replacing each numerical coefficient by unity, then (*, x+l, ..., x-\-rY=S (rtn) (1, 2, . . ., r)-V. ;-.0 \ J / "HDoura d'alg^bre sup^rieure, ed. 2, 1854, 324. «7Quar. Jour. Math., 5, 1862, 35-39. M8Proc. London Math. Soc., 20, 1889, 207. "•Zeitechrift Math. Phys., 34, 1889, 190-1. »°Proc. Edinburgh Math. Soc., 8, 1890, 16-19. ^Messenger Math., 22, 1892-3, 51-2; 23, 1893-4, 58. MSPrace Mat. Fiz., Warsaw, 7, 1896, 12-14 (PoUsh). MINouv. Ann. Math., (4), 5, 1905, 103. *<Nyt Tidsskrift for Mat., 21, B, 1909-10, 8-10. "•/Md., p. 36-7. SMMath. Quest, Educat. Times, 48, 1888, 115; (2), 22, 1912, 99; Amer. Math. Monthly, 22, 1915, 103, 138, 170. M7Introd. a la th^orie des nombres, 1862, 79-80, 17. »"Quar. Jour. Math., 7, 1866, 370-2. "•Giornale di Mat., 4, 1866, 344. Proof by Sharp, Math. Ques. Educ. Times, 47, 1887, 145-6; 63, 1895, 38. "'Ibid., 318-9. Cf. Wronski1" of Ch. VIII.