228 History of the Theory of Numbers. [Chap, viii 
which for g{x) = l yields the first formula of Lerch. Next, if the A;'s are 
primes and g is a prime distinct from them, 
2 Gix'^-q] k,,.. ., k,; x)= 2 (fc,-l, n; q)g{ki). 
x=0 /=1 
Finally, he treated f{x) of degree d = A-j — 2, whose constant term is prime to 
each A-, and coefficient of x''~' is divisible by the prime k^ if i<ks — k^. 
Gegenbauer^" noted that, if p — l—n is the rank of the system (3) 
modulo p, the congruence, satisfied by the distinct roots 5^0 of (2) and by 
these only, is given symboUcally by 
(-X--V 
\dai daj 
fli+fc I =0 (mod p) {%, k = 0,. . ., p-2). 
He obtained easily Kronecker's"^ form of the last congruence. He gave 
necessary and sufficient conditions, expressed in terms of a comphcated 
determinant and its /z — l successive derivatives with respect to Op_2, in 
order that (2) and a second congruence of degree p — 2 shall have jx common 
roots ?^0, and found the congruence satisfied by these ji common roots. 
He deduced determinantal expressions for the sum o-^ of the rth powers of 
the roots of (2), and for the coefficients in terms of the cr's. 
Michael Demeczky^^ would employ Euclid's process to find the g. c. d. 
G{x) modulo p of (2) and x'^—x. If G{x) = (mod p) is of degree v it has 
V real roots and these give all the real roots of (2) . Multiple roots are then 
treated. The case of any composite modulus is known to reduce to the 
case of p', p a prime. If (2) has X distinct real roots, not multiple roots, we 
can derive X real roots of /(a;) = (mod p'). If pi, . . . , p„ are distinct primes 
and if /(x) = (mod p,) has X, real roots, then/(x) = (mod pi. . .p„) has 
Xi. . .X„ real roots, and is satisfied by every integer x if the former are. 
Various sets of necessary and sufficient conditions are found that f{x) = 
(mod m=np'<) shall have m distinct real roots; one set is that/(x)=0 
(mod p'<) identically for each i. 
L. Gegenbauer^^ proved that a congruence modulo p, a prime, of degree 
p — 2 in each of n variables has a set of solutions each ^0 if and only if p 
divides the determinant of a cycUc matrix 
A« A' .. 
A'- 
-1 ' 
A"-- 
-2 
A^ 
A' A^ .. 
where A" is itself a cyclic matrix in B^,. . ., B'^~^; etc., until we reach 
matrices in the coefficients of the congruence. An upper limit is found for 
'"Sitzungsber. Ak. Wiss. Wien (Math.), 98, Ila, 1889, 652-72. 
"Math. u. Naturw. Berichte aus Ungarn, 8, 1889-90, 50-59. Math. 68 Term^s Ertesito, 7, 
1889, 131-8. 
»=Sitzungsber. Ak. Wiss. Wien (Math.), 99, Ila, 1890, 799-813. 
