100 HISTORY OF THE THEORY OP NUMBERS. [CHAP, m
S. Monteiro292 noted that 2n+l divides (2n)
J. Westlund293 reproduced the discussion by Serret256 and TchebycheL76
Glaisher294 proved his291 earlier theorems. Also, if p=2m+l is prime,
(fl&—Op$a(l» • • -j 2m)=/S2H-i(l, • • • > 27n) (mod p3) and, if t> 1, modulo p4. According as n is odd or even,
For m odd and >3, S2m-3(l,..., 2w-l) is divisible by m2, and £OT.2(12,..., -jm-1}2), S2m_4(l,..., 2m-1)
are divisible by m. He gave the values of Sr(l,..., n) and Ar—Sr(l9..., n—1) in terms of n for r=l,..., 7; the numerical values of Sr(l,..., n) for n^22, and a list of known theorems on the divisors of Ar and Sr. For r odd, 3^r^77i—2, S,(l,..., 2m—1) is divisible by m and, if w is a prime > 3, by m.2 He proved (ibid., p. 321) that, if l^r^ (p-3)/2, and Br is a Bernoulli number,
Glaisher295 gave the residues of <rk [Frost268] modulo p2 and p3 and proved that cr2, o-4). .., (Tp«3 are divisible by p, and cr3, <r5, . . ., crp_2 by p2, if p is a prime.
Glaisher296 proved that, if p is an odd prime,
Q 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 arithmetical progression.
F. Sibirani2960 proved for the Sn>m of Sylvester269 (designated sn,m+i) that
aMJornal Sc. Mat. Phys. e Nat., Lisbon, 5, 1898, 224. "3Proc. Indiana Ac. Sc., 1900, 103-4. "«Quar. Jour. Math., 31, 1900, 1-35. «5/bid., 329-39; 32, 1901, 271-305.