Chap. Ill] GENERALIZATIONS OF FeRMAt's THEOREM. 87 
The products of the various X^ by any one of them are congruent modulo k 
to the Xg in some order. Hence 
X/^w=ie^ (modifc), 
where R^ is the corresponding one of the 2' roots of x^^x (mod k). The 
analogous extension of Wilson's theorem is HXs^^Rs (mod k), the sign 
being minus only when k/a = p'', 2p' or 4 and at the same time a/s is odd. 
Here <r = np/^ if s = np,. Cf. Mitchell,^" Ch. V. 
F. RogeP^^ proved that, if p is a prime not dividing n, 
n-i = l+(f)(7i-l) + (|)(n-l)2+... + (f)(n-l)Hp, A: = ^, 
where p is divisible by every prime lying between k and p+l. 
Borel and Drach^^° investigated the most general polynominal in x divis- 
ible by m for all integral values of x, but not having all its coefficients 
divisible by m. If m = p''q^, . . . , where p, q,. . .are distinct primes, and if 
P{x), Q{x),. . . are the most general polynomials divisible by p", q^,. . ., 
respectively, that for m is evidently 
{P{x)+p'^f{x)\\Q{x)+q'g{x)\.... 
For a<p+l, the most general P{x) is proved to be 
iMx)Mx), Mx) =p''-\x^-x)\ 
where the/'s are arbitrary polynomials. For a<2(p+l), the most general 
P{x) is 
s/,<A,+ ste, rp,=ct>{x){x^-xy-Y-''-\ 
k=l k=l 
where 4>{x) = {x^—xy—p^~^{x^—x), and the/'s, g's are arbitrary poly- 
nomials. Note that ^^(x) -p'''-'^<f>lx) is divisible by p^'+^+K Cf. Nielsen.^^^ 
E. H. Moore^^^ proved the generalization of Fermat's theorem: 
Xi''"*-^ X p*""^ 
XiP 
Xi 
m p— 1 p— 1 
= n H . . . H {Xk+Ck+iXk+i+ . . . +c^xj (mod p). 
F. Gruber^^^ showed that, if n is composite and ai, . . . , a< are the ^=0(n) 
integers < n and prime to n, the congruence 
(1) x' — l = (x— fli). . .(x— a«) (mod n) 
is an identity in x if and only if n = 4 or 2p, where p is a prime 2*+l. 
"•Archiv Math. Phys., (2), 10, 1891, 84-94 (210). 
""Introduction th^orie des nombres, 1895, 339-342. 
">BuU. Amer. Math. Soc, 2, 1896, 189; cf. 13, 1906-7, 280. 
"»Math. Nat. Berichte aus Ungarn, 13, 1896, 413-7; Math, term^s ertesito, 14, 1896, 22-25. 
