102 History of the Theory of Numbers. [Chap. hi 
as k is or is not divisible by 4. For n even, S is divisible only by k/2 pro- 
vided n is not divisible by any prime factor, diminished by unity, of k. 
N. Nielsen^"^ wrote Cp for the sum of the products r at a time of 1, ... , 
p — 1, and 
s„(p)=2;s^ (7„(p)=S(-l)''-V. 
*-l «=1 
If p is a prime >2m+1, 
o-2n(p-l)=S2n(p-l)=0(modp), Sgn+iCp " 1) = O(niod p2). 
If p = 2n+l is a prime >3, and l^r^n — 1, Cf'^^ is divisible by p^. 
Nielsen^''^ proved that 2Di^'^^ is divisible by 2n for 2p+l^n, where D\ 
is the sum of the products of 1, 3, 5, . . . , 2n — 1 taken s at a time; also, 
2^"+'s2fl(n - 1) = 2^%,{2n - 1) (mod 4n^), 
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.29^ 
Nielsen^°^ proved the results last cited. Let p be an odd prime. If 
2n is not divisible by p — 1, S2r.(p — 1) = (mod p), S2„+i(p — 1)=0 (modp^). 
But if 2n is divisible by p — 1, 
S2n(p-1)=-1, S2„+i(p-l) = (modp), Sp(p - 1) = (mod p^). 
T. E. Mason^"^ proved that, if p is an odd prime and i an odd integer > 1, 
the sum Ai of the products i at a time of 1, . . . , p — 1 is divisible by p^. If 
p is a prime >3, Sk is divisible by p^ when k is odd and not of the form 
m(p — 1) + 1, by p when k is even and not of the form 7n{p — l), and not 
by p if A: is of the latter form. If A; = 7n(p — 1) + 1, s^ is divisible by p^ 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. Putnam^"^ proved Glaisher's^^^ theorem that s_„ is divisible by 
p if n is not a multiple of p — 1 , and 
(p-l)/2 9 — 9P 
2 jp-2=f_A(inodp). 
y-i p 
W. Meissner^^° arranged the residues modulo p, a prime, of the successive 
•o'K. Danske Vidensk. Selsk. Skrifter, (7), 10, 1913, 353. 
"•Annali di Mat., (3), 22, 1914, 81-94. 
•"Ann. sc. I'^cole norm, sup., (3), 31, 1914, 165, 196-7. 
"*T6hoku Math. Jour., 5, 1914, 136-141. 
»"Amer. Math. Monthly, 21, 1914, 220-2. 
"•Mitt. Math. Gesell. Hamburg, 5, 1915, 159-182. 
1 
i 
