258 History of the Theory of Numbers. [Chap, viil 
remain arbitrary, and that ^ is a function of them and one of the n's, which 
has an arbitrary rational value. 
A. Cauchy^" noted that if / and F are polynomials in x, Lagrange's 
interpolation formula leads to polynomials u and v such that uJ-\-vF = R, 
where i? 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 di\-iding both / and F. For /= x^—x, we may express i2 as a prod- 
uct of trigonometric functions. If also F{x)= (x"+l)/(a:+l), where n and 
p are primes, R=0 or ±2 according as p is or is not of the form nx-\-\. 
Hence the latter primes are the only ones dividing x^+l, but not x-\-\. 
Cauchy^^^ proved that a congruence /(x) = (mod p) of degree m<p is 
equivalent to (x— r)'</)(x) = 0, where 4> is of degree m—i, if and only if 
/(r)^0, /'(r) = 0,. . ., r-'\r)^Q (mod p), 
where p is a prime. The theorem fails if m'^p. He gave the method of 
Libri (M^moires, I) for solving the problem: Given"/(a:) = (mod p) of 
degree m^p and with exactly m roots, and/i(x) of degree l^m, to find a 
polynomial </)(a;), also with integral coeflficients, 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. 
Cauchy^^ stated that if 7 is an arbitrary modulus and if ri, . . ., r„, are 
roots of /(x)=0 (mod 7) such that each difference Vi—Tj is prime to 7, then 
f{x)={x-ri) . . .(x-rJQ(x) (mod 7). 
If in addition, m exceeds the degree of /(x), then/(x) = (mod 7) for every x. 
A congruence of degree n modulo p^, where p is a prime, has at most n 
roots unless every integer is a root. If /(r) = (mod 7) and if in the irre- 
ducible fraction equal to 
_ /(r) 
the denominator is prime to 7, then r— r7 is a root of /(a:)=0 (mod P). 
V. A. Lebesgue^^^ wrote a/b=c (mod p) if h is prime to p and a=bc 
(mod p), and a/b=c/d (mod p) \i h, d are prime to p and ad=bc (mod p). 
J. A. Serret^^^ stated and A. Genocchi proved that, if p is a prime, the 
sum of the mth. powers of the p" polynomials in x, of degree n — 1 and with 
integral coefficients <p, is a multiple of p if m<p'' — 1, but not if m = p'' — 1. 
J. A. Serret^^^ noted that all the real roots of a congruence f{x) = 
(mod p), where p is a prime, satisfy \j/{x)^0, where \f/ is the g. c. d. of f{x) 
andxP~^-l. 
'"Exercices 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. j 
i"Compte8 Rendus Paris, 25, 1847, 37; Oeuvres, (1), 10, 324-30. ~\ 
'"Nouv. Ann. Math., 9, 1850, 436. 
i^Nouv. Ann. Math., 13. 1854, 314; 14, 1855, 241-5 
"'Cours d'alg^bre sup6rieure, ed. 2, 1854, 321-3. 
