Chap. VIIIJ NuMBER OF RoOTS OF CONGRUENCES. 231 
L. Gegenbauer'*" proved that (2) has as a root a quadratic residue or 
non-residue of the prime p if and only if the respective determinant 
P = \ a^+i+o^+i+x \, N = \ a^+i-a^+i+, \ (i, /i = 0, . . ., tt-I) 
be divisible by p, where 7r= (p — 1)/2. From this it is proved that (2) has 
exactly ir—r distinct quadratic residues (or non-residues) of p as roots if 
and only if P (or N) and its tt — 1— r successive derivatives with respect to 
a,_i+ap_2 have the factor p, while the derivative of order tt— r is prime 
to p. These residues satisfy the congruence 
where K = P or N, while the j^th power of the sign of differentiation repre- 
sents the ^'th derivative. A second set of conditions is obtained. Con- 
gruence (2) has exactly tt — I—k distinct quadratic residues as roots if and 
only if the determinants of type P with now i = 0, . . ., k, k+1 and fi = 0,. . . , 
K, T, are divisible by p for r = /c+l, . . ., tt — 1; while p is not a factor of the 
determinant of type P with now i, fx = 0,. . ., k. These residues are the 
roots of 
«■ 
S I a^+i+a^+i+^ I x'~^~^=0 (mod p), 
T=(C 
where ^ = 0, . . ., k, and /x = 0, . . ., k — 1, t in the determinants. For non- 
residues We have only to use the differences of a's in place of sums. 
S. O. Satunovskij^^ noted that, for a prime modulus p, a congruence of 
degree n (n<p) has n distinct roots if and only if its discriminant is not 
divisible by p and Sp+q=S g+i (mod p) ior q = l,. . ., n — 1, where Sk is the 
sum of the kth powers of the n roots. 
A. Hurwitz^^ gave an expression for the number A^ of real roots of 
f{x)=aQ-\-aiX-\- . . . -\-arX''=0 (mod p), 
where p is a prime. By Fermat's theorem, 
-1 
N=X \l-f{xy-'\ (modp). 
a;=l 
Letf{xy-'^ = Co-\-CiX+ .... Then N is determined by 
Ar+l=Co+Cp_i+C2(,-i)+ . . . (mod p). 
Letf(xi, X2) be the homogeneous form of f(x). Let A be the number of 
sets of solutions of f{xi, a:2) = (mod p), regarding {xi, X2) and (x/, X2) as 
the same solution if Xi=pxi, X2=pX2 (mod p) for an integer p. Then 
A-l= -0^-1-0^71+2^^:1^1 ao'^o. . Mr^r (mod p), 
tto ! . . . a^ ! 
"Sitzungsber. Ak. Wiss. Wien (Math.), 110, Ila, 1901, 140-7. 
"Kazani Izv. fiz. mat. Obsc. (Math. Soc. Kasan), (2), 12, 1902, No. 3, 33-49. Zap. mat. otd. 
Obsc, 20, 1902, I-II. 
"Archiv Math. Phys., (3), 5, 1903, 17-27. 
