Chap. VIII] MISCELLANEOUS RESULTS ON CONGRUENCES. 259 
N. H. AbeP^^ proved that we can solve by radicals any abelian equation, 
i. e., one whose roots are r, 0(r), 0^(r) = <t)[<i){r)], . . ., where </> is a rational 
function. H. J. S. Smith^^^ concluded that when the roots of a congru- 
ence can be similarly expressed modulo p, its solution can evidently be 
reduced to the solution of binomial congruences, and the expressions for 
the roots of the corresponding equation may be interpreted as the roots 
of the congruence. For the special case a:"=l, this was done by Poinsot 
in 1813-20 in papers discussed in the chapter on primitive roots. 
M. Jenkins^^^" noted that all solutions of a^=l(mod x) are x= Un=UiU2 
. . .Un, where Ui is any divisor of any power of a — 1; u^ any divisor prime 
to a — 1, of any power of a'" — 1;. . .; u,, any divisor, prime to a^"-2 — 1, 
of any power of a^^-'^ — l. For a*+l = (mod x), modify the preceding 
by taking odd factors of a+1 instead of factors of a — 1. 
J. J. Sylvester^®° proved that if p is a prime and the congruence /(a:) = 
(mod p) of degree n has n real roots and if the resultant of f{x) and g{x) 
is divisible by p, then g{x)^0 has at least one root in common with /(a;) = 0. 
There are exactly p — 1 real roots of x^~^=l (mod p^). 
A. S. Hathaway^^^ noted the known similarity between equations and 
congruences for a prime modulus. He^^^ made abstruse remarks on higher 
congruences. 
G. Frattini^^^ proved that x^ — Dy'^=\ and x'^ — Dy^=\ are each solvable 
when the modulus is a prime p>5 and Dp^O. If d = B^—AC^O, then 
Ax'^-i-2Bx^y+Cy"=\ (mod p) is solvable since dx'^+XC can be made con- 
gruent to a square and hence to {Cy-{-Bx^y. Likewise for ax'^-\-2bx-\-c=y'^. 
A. Hurwitz^^^ discussed the congruence of fractions and the theory of 
the congruence of infinite series. If (/)(x) =ro-\-riX-{- . . . +r„a:V^-+ • • • and 
if yp{x) is a similar series with the coefficients s^, then and \l/ are called 
congruent modulo m if and only if Vn^s^ (mod m) for n = 1, 2, . . . . 
G. Cordone^^^ treated the general quartic congruence for a prime 
modulus ji by means of a cubic resolvent. The method is similar to Euler's 
solution of a quartic equation as presented by Giudice in Peano's Rivista 
di Matematica, vol. 2. For the special case x'^-\-%Hx^-\-K=0 (mod /x), 
set t = {ix — l)/2, r^ = 9H^ — K; then if K is a quadratic residue of /jl, there 
are four rational roots or none according as ( — 3/f+r)'=+l or not; but 
if K is a non-residue, there are two rational roots or none according as one 
of the congruences 
(-3H+r)'=4-l, i-ZH-ry^-l 
is satisfied or not. 
i^sjour. fur Math., 4, 1829, 131; Oeuvres, 1, 114. 
"sReport British Assoc. 1860, 120 seq., §66: Coll. M. Papers, 1, 141-5. 
"9aMath. Quest. Educ. Times, 6, 1866, 91-3. 
""Amer. Jour. Math., 2, 1879, 360-1; Johns Hopkins University Circulars, 1, 1881, 131. Coll. 
Papers, 3, 320-1. 
i"Johns Hopkins Univ. Circulars, 1, 1881, 97. "^Amer. Jour. Math., 6, 1884, 316-330. 
"^Rendiconti Reale Accad. Lincei, Rome, (4), 1, 1885, 140-2. 
i"Acta Mathematica, 19, 1895, 356. 
«6Rendiconti Circolo Mat. di Palermo 9, 1895, 209-243. 
