Chap. VIII] NuMBER OF RoOTS OF CONGRUENCES. 225 
Denote by P the sum of the first / terms of F and by Q the sum of the last 
k—f terms. Let gr be a primitive root of p. Let P° be the number of sets 
of solutions of P=0 (mod p); P^'^ the number for P^g^ (mod p); Q^ and 
Q^'^ the corresponding numbers for Q=0, Q=g'. Then the number of 
sets of solution of P=Q (mod p) is P°Q°+/iS:=rP^*^Q^*\ Hence we may 
deduce the number of sets of solutions of P=0 from the numbers for 
P = Aand Q= -A. For P= a, we employ P = P, Q = /x'"and get P° = P° 
-\-{'p — l)P^^\ which determines the desired P''^\ 
The theory is applied in detail to (1) for m = 2, k arbitrary, and for 
w = 3, 4, k = 2. Finally, the method of Libri^^ is amplified. 
Th. Schonemann^^ noted that, if Sk is the sum of the A;th powers of the 
roots of an equation x"+ . . . =0 with integral coefficients, that of x"" being 
unity, and if >S(p_i)«=n (mod p) for <= 1, 2, . . ., w, where p is a prime >n, 
the corresponding congruence a;"+ . . . = (mod p) has n real roots. 
A. L. Cauchy^^ considered F{x) = Q (mod M), with M=AB. . ., where 
A, B,. . . are powers of distinct primes. If F{x) = (mod A) has a roots, 
F(x) = (mod B) has /S roots, etc., the proposed congruence has a/3. . . roots 
in all. For, if a, 6, . . . are roots for the moduli A, B,. . . and X=a (mod A), 
X=b (mod P), . . . , then X is a root for modulus M. 
P. L. Tchebychef^° proved that, if p is a prime, a congruence /(x)=0 
(mod p) of degree m<p has m roots if and only if the coefficients of the 
remainder obtained by dividing x^—x hj f{x) are all divisible by p. 
Ch. Hermite^^ proved the theorem: If fx and jjl' are the numbers of 
sets of solutions of 4>{x, y)=0 for the respective moduli M and M', which 
are relatively prime, the number of sets of solutions modulo MM' is /jl/j,'. 
If 0=0 is solvable for a prime modulus p, it will be solvable modulo p" if 
have no common sets of solutions. In this case, the number of sets of 
solutions modulo p" is p"~V if tt is the number for modulus p. Similar 
results are said to hold for any number k of unknowns. If ikf is a product 
of powers of the distinct primes pi, . . ., p„, and if tt, is the number of sets 
of solutions of the congruence modulo Pi, then the number of sets for 
modulus M is 
M' 
k~l TTi-.-TTn 
For a:^+A|/^=A (mod M), we have Xj = p, — (— A/p»), where (a/p) is 
=«= 1 according as a is a quadratic residue or non-residue of p. 
JuUus 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.^^ 
"Jour, fiir Math., 19, 1839, 293. 
"Comptes Rendus Paris, 25, 1847, 36; Oeuvres, (1), 10, 324. 
"Theorie der Congruenzen, in Russian, 1849; in German, 1889, §21. 
"Jour, fiir Math., 47, 1854, 351-7; Oeuvres, 1, 243-250. 
