CHAP. Ill] GENERALIZATIONS OP FEBMAT'S THEOREM. 87 The products of the various -X", by any one of them are congruent modulo k to the Xa in some order. Hence XJ*(k)=Rs (mod*), where R8 is the corresponding one of the 2l roots of x2^x (mod k). The analogous extension of Wilson's theorem is HXa==*=Rs (mod k), the sign being minus only when k/<r = pv, 2pv or 4 and at the same time cr/s is odd. Here a - Up? if s = lip,, Cf . Mitchell,50 Ch. V. F. Rogel179 proved that, if p is a prime not dividing n, where p is divisible by every prime lying between k and p+1. Borel and Drach180 investigated the most general polynominal in x divisible by m for all integral values of x} but not having all its coefficients divisible by m. If m = p°-q^y . . . , where p, q, . . .are distinct primes, and if P(x)j Q(x)}. . . are the most general polynomials divisible by pa} q*3, . . ., respectively, that for m is evidently \P(x)+p?(x)\\Q(x)+q>g(x)\.... For o<p+l, the most general P(x) is proved to be where the/'s are arbitrary polynomials. For a<2(p+l), the most general P(s) is where <^(x) = (a;p— x)p— pp"1(rcp— x), and the /;s, ^;s are arbitrary polynomials. Note that </>p(z) -p^l^>(x) is divisible by p^^1. Cf. Nielsen.194 E. H. Moore181 proved the generalization of Format's theorem: pm-l pm-l • • ' m m p-l p-l n n ... n (xk+ck+lxk+i+ . . . +cmxm) (mod p). F. Gruber182 showed that, if n is composite and ab . . . , a< are the t=4>(n) integers <n and prime to n, the congruence (1) a;' — Iz=-(x— ai). . .(x-^) (mod n) is an identity in a; if and only if n = 4 or 2p, where /> is a prime 2*-|-l. "•Archiv Math. Phys., (2), 10, 1891, 84-94 (210). ""Introduction th6oric des nombrea, 1895, 339-342. "lBull. Amer. Math. Soc., 2, 1896, 189; cf. 13, 1906-7, 280. »»Math. Nat. Berichte aua Ungara, 13, 1896, 413-7; Math, term&j ertcsito, 14, 1896, 22-25.