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CHAP, viii]           NUMBER OF ROOTS OF CONGRUENCES.                      231
L. Gegenbauer40 proved that (2) has as a root a quadratic residue or non-residue of the prime p if and only if the respective determinant
be divisible by p, where TT= (p — 1)/2. From this it is proved that (2) has exactly TT — 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 <V-i+<V-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 *>th power of the sign of differentiation represents the vth derivative. A second set of conditions is obtained. Congruence (2) has exactly TT — 1— K distinct quadratic residues as roots if and only if the determinants of type P with now i = 0, . . . , K, K+l and /j, = 0, . . . , K, r, are divisible by p for r = ic-t-l, . . ., TT— 1; while p is not a factor of the determinant of type P with now i, Ai = 0, . . ., K. These residues are the roots of
£   I aM+i+aM+i+, | s-^'sO (mod p),
where i = 0, . . ., K, and ^ = 0, . . ., K — 1, r in the determinants. For non-residues we have only to use the differences of a's in place of sums.
S. O. Satunovskij41 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=SQ+i (mod p) for q = l,. . ., n — 1, where Sk is the sum of the /cth powers of the n roots.
A. Hurwitz42 gave an expression for the number N of real roots of
f(x)=aQ+a1x+ . . . + arxr=0 (mod p), where p is a prime.    By Format's theorem,
N=*2 \l-f(x)*~l\ (modp).
Let f(x)p~l = C0+Clx+....    Then N is determined by
Ar+l=C0+Cp_1+C2(p.1)+... (modp). i, £2) be the homogeneous form of f(x).   Let A be the number of
sets of solutions of f(xi, £2) = 0 (mod p), regarding (xlf x2} and (x/, £2') as the same solution if Xi=pXi, x2'^px2 (mod p) for an integer p.    Then
l             .
a0I. . .ar!
40Sitzungsber. Ak. Wiss. Wien (Math.), 110, Ila, 1901, 140-7.
"Kazan! 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.