100 
History of the Theory of Numbers. 
[Chap. Ill 
S. Monteiro'^- noted that 2n+l divides {2n)\Z\n/r. 
J. Westlund-^^ reproduced the discussion by Serret^^^ and Tchebychef.'^^ 
Glaisher^^ proved his^^^ earlier theorems. Also, ii p = 27n+l is prime, 
{m-t)pS2t{l,.. ., 2m)=S2t+i{l,. ■ ., 2m) (mod f) 
and, if i>l, modulo p'^. According as n is odd or even, 
>S2t(l,. . ., n)=S2t{l,. . ., n-1) (mod n^ or ^n^). 
For m odd and >3, S2m-z0-i- . ., 2m — 1) is divisible by m^, and 
^„_2(1^..., \m-l\^), ^2n.-4(l,...,2m-l) 
are divisible by m. He gave the values of Sr{\,. ■ ■, n) and Ar = Sr{l,. . ., 
n — 1) in terms of n for r = l,. . ., 7; the numerical values of 5^(1,. . ., n) 
for n^22, and a list of known theorems on the divisors of Ar and Sr. For 
r odd, 3^r^m — 2, Sr{l, . . ., 2w— 1) is divisible by m and, if w is a prime 
>3, by m.^ He proved {ibid., p. 321) that, if l^r^ (p-3)/2, and 5, is a 
BernoulU number, 
2.S2.+i(l,. . ., p-l)_-{2r+l)S2r{l,..., p-1) 
V V 
^2.(i,...,p-i)_(-ir5: 
V 
2t 
(modp). 
Glaisher^^^ gave the residues of a^ [Frost^^^] modulo p^ and p^ and proved 
that (72, o'i, •• . , o-p-3 are divisible by p, and 0-3, cs, . . . , o-p_2 by p^, if p is 
a prime. 
Glaisher^^® proved that, if p is an odd prime. 
■'•"' o2n I r2n I ' 
(p-2) 
2n 
^0 or — I (mod p), 
according as 2n is not or is a multiple of p — 1. He obtained (pp. 154-162) 
the residue of the sum of the inverses of like powers of numbers in arith- 
metical progression. 
F. Sibirani^^^" proved for the Sn,m of Sylvester^^^ (designated Sn,m-\-\) that 
^n,n '^n— l,n • • • *^n— Jfe+l.n 
On+A;— l,n+*— 1 Sn+k—2,n+k—l ■ ■ ■ ^n.n+k—l 
t 
= inl)K 
"'Jornal Sc. Mat. Phys. e Nat., Lisbon, 5, 1898, 224. 
»»Proc. Indiana Ac. Sc, 1900, 103-4, 
"^Quar. Jour. Math., 31, 1900, 1-35. 
»'/Wd., 329-39; 32, 1901, 271-305. 
"•Messenger Math., 30, 1900-1, 26-31. 
'wPeriodico di Mat., 16, 1900-1, 279-284. 
