88 HlSTOBY OP THE THEOEY OF NUMBEBS. [CHAP. Ill E. Malo183 employed integers A/ and set u=xf*z, un~ldu Since ?, Q~- , 2m+n,...), P where k takes the values n, m+n,.. .which are ^pfp. If no prime factor of such a k occurs in the denominator of the expansion of up/p, the latter is an integer; this is the case if p is a prime and ju^ 2. For m = n = l, M=2, *-«>•-©-©«+...«* °-3=t*°-2' we get ojp=ap—a and hence Fermat's theorem. L. Kronecker184 generalized Fermat's and Wilson's theorems to modular systems. R. Le Vavasseur185 obtained a result evidently equivalent to that by Moore181 for the non-homogeneous case xm=L M. Bauer186 proved that if n^p'm, where m is not divisible by the odd prune p, and aly..., at are the t—<t>(ri) integers <n and prime to n, identically in x. If p = 2 and x> 1, the product is identically congruent to (a;2 — I)'72. Hence he found the values of d, n for which (1) holds modulo d, when d is a divisor of n. If p denotes an odd prime and q a prime 2l'+1, the values are n 20 4 p«,2p- 2%2'Ma... M. Bauer187 determined how n and JV" must be chosen so that xn— 1 shall be congruent modulo AT to a product of linear functions. We may restrict N to the case of a power of a prime. If p is an odd prime, xn — I is congruent modulo pa to. a product of linear functions only when p^l (mod n), a arbitrary, or when n = prm, a = l, p=l (mod w). For p = 2, only when n = 2^, a = l, or n = 2, a arbitrary. For the case n a prime, the problem was treated otherwise by Perott.188 M. Bauer189 noted that, if n—prm, where m is not divisible by the odd prime p, U(x-i) = (xp-x)n/p (modp*). h., 7, 1900, 281, 312. 1MVorlesungen uber Zahlentheorie, I, 1901, 167, 192, 220-2. l«Comptes Rendus Paris, 135, 1902, 949; M4m. Ac. Sc. Toulouse, (10), 3, 1903, 39-48. 1MNouv. Ann. Math., (4), 2, 1902, 256-264. 187Math. Nat. Berichte aus Ungarn, 20, 1902, 34-38; Math. 6s Phys. Lapok, 10, 1901, 274-8 (pp. 14&-152 relate to the "theory of Format's congruence": no report is available). 18«Amer. Jour. Math., 11, 1888; 13, 1891. "•Math. 6s Phys. Lapok, 12, 1903, 159-160.