96 History of the Theory of Numbers. [Chap, hi 
J. A. Serret^^^ 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. Wolstenholme^" proved that the numerators of 
1+UU...+ 1 1+1 + . ■ 1 
2 ' 3 ' •■• ' p-r - . 22 ' ••• ' (p-l)2 
are divisible by p^ and p respectively, if p is a prime >3. Proofs have also 
been given by C. Leudesdorf^^s, A. Rieke,^^^ E. Allardice,^^" G. Osborn,^" 
L. Birkenmajer,^" P. Niewenglowski,^^ N. Nielsen,^" H. Valentiner,^®^ 
and others.^^^ 
V. A. Lebesgue^®' proved that s^ is divisible by p if w is not divisible 
by p — 1 by use of the identities 
(n+1) S k{k+l) . . .ik-\-n-l)=xix+l) . . .(x+n) (n = l,. . ., p-1). 
k=l 
P. Frost^^^ proved that, if p is a prime not dividing 2^'' — 1, the numera- 
tors of a2r, (T2r-i, p(2r — l)o-2r+2(T2r-i are divisible by p, p^, p^, respectively, 
where 
1J_1_L 1 
2* ' •" ' (p-1)' 
The numerator of the sum of the first half of the terms of 0*2, is divisible by 
p; likewise that of the sum of the odd terms. 
J. J. Sylvester^^^ stated that the sum S^, m of all products of n distinct 
numbers chosen from 1,. . ., m is the coefficient of T in the expansion of 
{l+t){l-\-2t) . . . (1+wO and is divisible by each prime >n+l contained in 
any term of the set m— n+l,. . ., m, m+1. 
E. Fergola"" stated that, if (a, 6, . . . , ly represents the expression 
obtained from the expansion of (a+6H- . . . +0" by replacing each numerical 
coefficient by unity, then 
(X, x+1,. . ., x+rr= i CY)^^' 2,. . ., rr-^x^. 
*"Coiirs d'algSbre sup^rieure, ed. 2, 1854, 324. 
«'Quar. Jour. Math., 5, 1862, 35-39. 
"sProc. London Math. Soc, 20, 1889, 207. 
«»Zeit8chrift Math. Phys., 34, 1889, 190-1. 
««oProc. Edinburgh Math. Soc, 8, 1890, 16-19. 
"'Messenger Math., 22, 1892-3, 51-2; 23, 1893-4, 58. 
"^Prace Mat. Fiz., Warsaw, 7, 1896, 12-14 (Polish). 
M'Nouv. Ann. Math., (4), 5, 1905, 103. 
««Nyt Tidsskrift for Mat., 21, B, 1909-10, 8-10. 
^Ibid., p. 36-7. 
^^Math. Quest, Educat. Times, 48, 1888, 115; (2), 22, 1912, 99; Amer. Math. Monthly, 22, 1915, 
103, 138, 170. 
«^Introd. k la thdorie des nombres, 1862, 79-80, 17. 
»8Quar. Jour. Math., 7, 1866, 370-2. 
M»Giomale di Mat., 4, 1866, 344. Proof by Sharp, Math. Ques. Educ. Times, 47, 1887, 145-6; 
63, 1895, 38. 
"oibid., 318-9. Cf. Wronski"! of Ch. VIII. 
