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102                 HISTORY OF THE THEORY OF NUMBERS.            [CHAP, in
as k is or is not divisible by 4.   For n even, S is divisible only by k/2 provided n is not divisible by any prime factor, diminished by unity, of k.
N, Nielsen305 wrote  for the sum of the products r at a time of 1, . . . , p  1, and
If p is a prune >2n+l,
If p=2n+l is a prime >3, and l^r^nl, Cpr+1 is divisible by p2.
Nielsen306 proved that 2DJ+1 is divisible by 2n for 2p+l^n, where Di is the sum of the products of 1, 3, 5, . . . , 2n 1 taken s at a time; also,
22fl+1s2fl(n -l)s 22's2fl(2n - 1)       (mod 4n2) ,
and analogous congruences between sums of powers of successive even or successive odd integers, also when alternate terms are negative. He proved (pp. 258-260) relations between the C;s, including the final formulas by Glaisher,294
Nielsen307 proved the results last cited. Let p be an odd prime. If 2n is not divisible by p  1, $2(p--l)=0 (mod p), s2n+i(p-l)^0 (modp2). But if 2n is divisible by p 1,
l)=0 (modp),   sp(p-l)sO (modp2).
T. E. Mason308 proved that, if p is an odd prime and i an odd integer > 1, the sum A< of the products i at a time of 1, . . . , p  1 is divisible by p2. If p is a prime > 3, s* is divisible by p2 when k is odd and not of the form m(pl)+l, by p when k is even and not of the form ra(p  1), and not by p if k is of the latter form. If k = m(p  1) + 1, sk is divisible by p2 or p according as k is or is not divisible by p. Let p be composite and r its least prime factor; then r  1 is the least integer t for which At is not divisible by p and conversely. Hence p is a prime if and only if p  1 is the least t for which At is not divisible by p. The last two theorems hold also if we replace A's by s's.
T. M. Putnam309 proved Glaisher's296 theorem that $_ is divisible by p if n is not a multiple of p  1, and
(p-D/2          2  2P
j-i               p
W. Meissner310 arranged the residues modulo p, a prime, of the successive
I05K. Danske Vidensk. Selsk. Skrifter, (7), 10, 1913, 353.
IMAnnali di Mat., (3), 22, 1914, 81-94.
7Ann. BC. 1'gcole norm, sup., (3), 31, 1914, 165, 196-7.
8T6hoku Math. Jour., 5, 1914, 136-141.
109Amer. Math. Monthly, 21, 1914, 220-2.
310Mitt. Math. Gesell. Hamburg, 5, 1915, 159-182.