History Of The Theory Of Numbers - I

CHAP. VIII] NtJMBEK OF ROOTS OF CONGRUENCES. 229
the number of sets of solutions each not divisible by p. He proved that
. 2Z1 n
S ajXj 2 + 2 a9+jxt has p"*'-1 sets of solutions. Of these,
. 2Z1 n
S ajXj 2 + 2 a9+jxt+j+bz~Q (mod p)
have each a^O, where r is the number of the 2* integers
which are divisible by p. The number of sets of solutions of
2Z1 n
S £*#, 2 + S <Vwavw+tesO (mod p) y— i j-i
is expressed in terms of the functions used for quadratic congruences.
*E. Snopek38 gave a generalization of Konig's criterion for the solvability of a congruence modulo p.
L. Gegenbauer34 proved that if the p congruences
"s2 3a0p-2-*s 0 (mod p) (X = 0, 1, . . . , p - 1)
jt»o
have in common at least p — p distinct roots not divisible by p then all p-rowed determinants in the matrix (zk\) are divisible by p. The converse is proved when a certain condition holds. By specialization, Konig's theorem is obtained.
Gegenbauer86 proved that, if r is less than the prune p and if ZQ, . . . , zr_i are incongruent and not divisible by p, the system of linear congruences
(4) ^ &n.,y4sO (mod p) (p = 0, 1,. . ., p-2)
Jt=0
has all its sets of solutions of the form
(5) y^tA? (t = 0,l,...,p-2)
x-o
or not, according as the matrix (bk+p), k=r, r+1, . . ., p — 2; p = 0, . . ., p— 2, has a p— r— 1 rowed determinant prime to p or not. Next, if
(6) PS bkxk=Q (mod p)
fc-O
has exactly r distinct roots ZQ, . . . , zr~i each not divisible by p, every system of solutions of (4) is given by (5), and conversely. By combining this theorem of Kronecker's with the former, we obtain Kronecker's form of Konig's theorem.
"Prace Mat. Fiz., Warsaw, 4, 1893, 63-70 (in Polish). "Sitzungsber. Ak. Wiss. Wien (Math.), 102, Ha, 1893, 549-64. "Monatshefte Math. Phys., 5, 1894, 230-2. Cf. Gegenbauer.*'