78 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in Take each e#=l; then N=ap since the number of the specified combinations becomes the sum of all products of p factors unity, one' from each row of the table. Thus R. W. Genese108 proved Euler's theorem essentially as did Laisant.91 M. F. Daniels109 proved the generalized Wilson theorem. If \f/(n) denotes the product of the integers <n and prime to n, he proved by induction that ^(p*) = -l (mod pw) for p an"bdd prime. For, if px, . . ., pn are the integers <p* and prime to it, then Pi+jp', . . -yPn+JP'ti^Q* 1, . . .,p— 1) are the integers <pir+1 and prime to it. He proved similarly by induction that M211) = + 1 (mod 2') if TT> 2. Evidently ^(2) ss 1 (mod 2) , ^(4) = - 1 (mod 4). If m-cftf . . . and w = Zx, where I is a new prime, then ^(m)=e (mod m), ^(n)=r? (mod n) lead by the preceding method to ^(mn)=€v>(n) (mod Ttt), viz., 1, unless n = 2. The theorem now follows easily. E. Lucas110 noted that, if re is prime to n=AB. . ., where A, B,. . . are powers of distinct primes, and if <£ is the 1. c. m. of <t>(A), </>(-#)>• • •> then £*= 1 (mod ft) . In case A = 2k, k> 2, we may replace <t>(A) by its half. To get a congruence holding whether or not x is prime to n, multiply the former congruence by xff, where a- is the greatest exponent of the prime factors of n. Note that 4>+<r<n [Bachmann129' 143]. Carmichael139 wrote X(n) for <j>. E. Lucas111 found A*"1^"1 in two ways by the theory of differences. Equating the two results, we have Each power on the right is =1 (mod p). Thus (p-l)!s(i-i)p-i-is-l (modp). P. A. MacMahon112 proved Fermat's theorem by showing that the number of circular permutations of p distinct things n at a time, repetitions allowed, is where d ranges over the divisors of n. For n a prime, this gives pn-h(n-l)p^0, pn^p (mod n). Another specialization led to Euler's generalization. E. Maillet113 applied Sylow's theorem on subgroups whose order is the highest power ph of a prime p dividing the order w of a group, viz., 108British Association Report, 1888, 580-1. 108Lineaire Congruences, Diss. Amsterdam, 1890, 104-114. 110BuU. Ac. So. St. P6tersbourg, 33, 1890, 496. mMathesis, (2), 1, 1891, 11; ThSorie des nombres, 1891, 432. U2Proc. London Math. Soc., 23, 1891-2, 305-313. U8Recherclies sur les substitutions, Th&se, Paris, 1892, 115.