s 
146 History of the Theory of Numbers. [Chap, v 
Hence if a ranges over the integers ^ n and prime to n, 
Z(a — nBy = or a multiple of mpp 
according as p is odd or even. By this recursion formula, 
L. Gegenbauer"^ gave a formula including those of Nazimov^^^ and 
Zsigmondy." For any functions xid), Xiid) , f i^i, ■ ■ ■ , x,), 
m /^\ /^\1 [m/d] 
f{KXi,. . ., /cx,)2x(5)xiM =^x(d)xA^) ^ 2 ^ fidKX^,. . ., dKX,), 
where d ranges over all divisors of n which have some definite property P, 
while 5 ranges over those common divisors of n, Xi,..., x, which have 
property P. Various special choices are made for x> Xi> / and P. For 
instance, property P may be that d is an exact pth power, whence, if p = 1, 
d is any divisor of n. The special results obtained relate mainly to new 
number-theoretic functions without great interest and suggested apparently 
by the topic in hand. 
T. del Beccaro^'^ noted that (t>k{n) is divisible by n if A; is odd [Binet^^^]. 
When n is a power of 2, 
l^+2*+...4-(n-l)* = Oor0(n) (modn), 
according as k is odd or even. His proof of (1) is due to Euler. 
J. W. L. Glaisher^^^ proved that, if a, h,. . . are any divisors of x such 
that their product is also a divisor, the sum of the nth. powers of the integers 
< X and not divisible by a or 6, . . . , is 
where s is the number of the divisors a, 6, ... , and 
li a,h,. . . are all the prime factors of x, this result becomes Thacker's.^^" 
N. Nielsen^^^ proved by induction on y that the sum of the nth powers 
of the positive integers <mM and prime to M = pi^. . .ply is 
"'""'^'^W+(-i)-'f'^-^'"' C^+iV (™m)"--' n (P---1). 
n+l «=in+l \ zs y ,=i 
The case m= 1 gives Thacker's^^° result. That result shows {ihid., p. 179) 
that 02n(w) and <^2n+i(^) are divisible by m and m^ respectively, for l^n 
^ (pi— 3)/2, where pi is the least prime factor of m, and also gives the resi- 
dues of the quotients modulo m. Corresponding theorems therefore hold 
for the sum of the products of the integers < m and prime to m, taken t at a. 
time. 
i"Sitzungsberichte Ak. Wiss. Wien (Math.), 102, 1893, Ila, 1265-94. 
"♦Atti R. Accad. Lined, Mem. CI. Fis. Mat., 1, 1894, 344-371. 
i«Messenger Math., 28, 1898-9, 39-41. 
"»Oversigt Danske Vidensk. Selsk. Forhandlinger, 1915, 509-12; cf. 178-9. 
