CHAP. VIII] NUMBEH OF ROOTS OF CONGKTTENCES. 225
Denote by P the sum of the first/ terms of F and by Q the sum of the last k—f terms. -Let g be a primitive root of p. Let P° be the number of sets-of solutions of P=0 (mod p); P(i) the number for P=gi (mod p); Q° and Q(i) the corresponding numbers for Qs=0, Q=gi. Then the number of sets of solution of P=Q (mod p) is P°Q°+hZ^P(i)Q(i). Hence we may deduce the number of sets of solutions of F=0 from the numbers for P=A and QE= -A. For F=s a, we employ P=F, Q = gkxm and get F°=P° + (p — l)P(fc), which determines the desired P(A).
The theory is applied in detail to (1) for w = 2, k arbitrary, and for w = 3, 4, fc = 2. Finally, the method of Libri16 is amplified.
Th. Schonemann18 noted that, if Sk is the sum of the kth powers of the roots of an equation xn+ . . . =0 with integral coefficients, that of xn being unity, and if /S(p_i),=w (mod p) for £=1, 2, . . ., n, where p is a prime >n, the corresponding congruence xn+ . . .=0 (mod p) has n real roots.
A. L. Cauchy19 considered ^(x)=0 (mod M), with M=AB. . ., where A, B,. . . are powers of distinct primes. If F(x)^Q (mod A) has a roots, F(x)=Q (mod B) has /3 roots, etc., the proposed congruence has a/3. . . roots in all. For, if a, b, ... are roots for the moduli A, B, . . . and X= a (mod A) , JT=6 (mod B), . . ., then JT is a root for modulus M.
P. L. Tchebychef 20 proved that, if p is a prime, a congruence /(#) = 0 (mod p) of degree m<p has m roots if and only if the coefficients of the remainder obtained by dividing xv— x by /(z) are all divisible by p.
Ch. Hermite21 proved the theorem: If /x and MX are the numbers of sets of solutions of <£(x, 2/)=0 for the respective moduli M and M1, which are relatively prime, the number of sets of solutions modulo MM' is /i/i'. If 0=0 is solvable for a prime modulus p, it will be solvable modulo pn if
have no common sets of solutions. In this case, the number of sets^ of solutions modulo pn is pn~lir if TT is the number for modulus p. Similar results are said to hold for any number k of unknowns. If M is a product of powers of the distinct primes plr . . ., pn, and if TT,- is the number of sets of solutions of the congruence modulo pit then the number of sets for modulus M is
For x2+Aj/2=A (mod M), we have Tri = pi— (— ^4/p»), where (a/p) is =±= 1 according as a is a quadratic residue or non-residue of p.
Julius Konig gave a theorem in a seminar at the Technische Hochschule in Budapest during the winter, 1881-2, which was published in the following paper and that by Rados.24
18Jour. far Math., 19, 1839, 293.
"Comptes Rendus Paris, 25, 1847, 36; Oeuvres, (1), 10, 324. a°Theorie der Congruenzen, in Russian, 1849; in German, 1889, §21. 11 Jour, fur Math., 47, 1854, 351-7; Oeuvres, 1, 243-250.