258 HISTORY OP THE THEORY OF NUMBERS. [CHAP, vm remain arbitrary, and that £ is a function of them and one of the n's, which has an arbitrary rational value. A. Cauchy162 noted that if / and F are polynomials in x, Lagrange's interpolation formula leads to polynomials u and v such that uf+vF=R, where R is a constant [provided / and F have no common factor]. If the coefficients are all integers, R is an integer. Hence R is the greatest of the integers dividing both/ and F. For /= xp—x, we may express R as a product of trigonometric functions. If also F(x)= (xn+l)/(x+I), where n and p are primes, R=0 or =*=2 according as p is or is not of the form nx+1. Hence the latter primes are the only ones dividing #n-fl, but not z-fl. Cauchy153 proved that a congruence/(x)=0 (mod p) of degree m<p is equivalent to (x—r)V(#)=0, where <t> is of degree m—i, if and only if /(r)s=0, /'(r)=0,..., /(*"1)(r)sO (mod p), where p is a prime. The theorem fails if m^p. He gave the method of Libri (M&noires, I) for solving the problem: Given f(x)z=Q (mod p) of degree m^p and with exactly m roots, and/i(o;) of degree l^m, to find a polynomial $(x), also with integral coefficients, whose roots are the roots common to/ and/i. He gave the usual theorem on the number of roots of a binomial congruence and noted conditions that a quartic congruence have four roots. Cauchy154 stated that if J is an arbitrary modulus and if rly..., rm are roots of/(x)=0 (mod /) such that each difference n—ry is prime to /, then /(z)= (z-n)... (x-rm)Q(x) (mod /). If in addition, m exceeds the degree of /(#), then f(x)=0 (mod 7) for every x. A congruence of degree n modulo px, where p is a prime, has at most n roots unless every integer is a root. If /(r) = 0 (mod J) and if in the irreducible fraction equal to the denominator is prime to /, then r— rl is a root of /(x) = 0 (mod 72). V. A. Lebesgue155 wrote a/&=c (mod p) if 6 is prime to p and a^bc (mod p), and a/b=c/d (mod p) if 6, d are prime to p and ad^bc (mod p). J. A. Serret156 stated and A. Genocchi proved that, if p is a prime, the sum of the mth powers of the pn polynomials in x, of degree n — 1 and with integral coefficients <p, is a multiple of p if m<pn— 1, but not if m = pn — l. J. A. Serret157 noted that all the real roots of a congruence /(x) = 0 (mod p), where p is a prime, satisfy ^(x)=0, where ^ is the g. c. d. of f(x) l62Exercices de Math., 1, 1826, 160-6; Bull. Soc. Philomatique; Oeuvres, (2), 6, 202-8. ^Exercices de Math., 4, 1829, 253-279; Oeuvres, (2), 9, 298-326. 1MComptes Rendus Paris, 25, 1847, 37; Oeuvres, (1), 10, 324-30. lwNouv. Ann. Math., 9, 1850, 436. lMNouv. Ann. Math., 13, 1854, 314; 14, 1855, 241-5 l67Coura d'algebre supe>ieure, ed. 2, 1854, 321-3.