88 HISTORY OF THE THEORY OF NUMBERS. [CHAP, n 



where < is Euler's function; finally, let /(ft, d) = <t>(d) if no prime occurs in 

 d to a higher power than in n, so that n is divisible by d. Then 



summed for all the divisors d of AT, where ()) is a binomial coefficient 

 and is zero if j is not an integer. By the second formula, f(n, M) equals 

 the difference between the numbers of representations of n by an odd and 

 by an even number of summands not divisible by the modulus M . 



Von Sterneck 232 proved that the number [ft]; of representations of n as 

 the residue modulo M of a sum of i elements chosen from 0, 1, -, M 1, 

 repetitions allowed, is 



- 1\ 



summed for all the divisors d of M . If the elements are chosen from the 

 numbers e\, - , e v incongruent modulo M, then 



- I)"' 1 Z - 



Von Sterneck 233 determined (TO),- and [n],- for a prime power modulus. 



O. E. Glenn 234 found the number of sets of solutions of X + M + *> = 

 (mod p 1) and ofX + ju + j> + = (mod p 1), the order of X, n, 

 - - being disregarded, and p being prime. 



D. N. Lehmer 235 proved that a l x l + + ^nX n + a n+i = (mod m) 

 has m n ~ l 8 solutions or no solution according as the g.c.d. 8 of a\ t , a n , m 

 does or does not divide a n+ i. 



L. Aubry 236 noted that if A is prime to N and if B/ ^N is not integral, 

 Ax = By (mod AT) is solvable in integers =j= numerically < 



SYSTEM OF LINEAR CONGRUENCES. 



A. M. Legendre 237 treated the problem to find integers x such that, if 

 a and 6, a' and 6', are relatively prime, 



C tZ *C C 



6 6' 



are all integers. The first condition gives x = m + bz. Then the second 

 condition requires that a'bz + a'm c' be divisible by &', which is im- 



232 Sitzungsber. Akad. Wiss. Wien (Math.), 114, Ha, 1905, 711-730. 



233 /fod., 118, Ha, 1909, 119-132. 



234 Amer. Math. Monthly, 13, 1906, 59-60, 112-4. 



Ibid., 20, 1913, 155-6. 



136 Mathesis, (4), 3, 1913, 33-5. 



237 Th6orie des nombres, 1798, 33; cd. 2, 1805, 25; ed. 3, 1830, I, 29; Maser, I, p. 29. 



