78 History of the Theory of Numbers. [Chap, hi 
Take each e^ = 1 ; then N = a^ smce the number of the specified combina- 
tions becomes the sum of all products of p factors unity, one from each row 
of the table. Thus 
0"= ( ^j=a (mod p). 
p-i 
R. W. Genese^°^ proved Euler's theorem essentially as did Laisant.'^ 
M. F. Daniels^^^ proved the generahzed Wilson theorem. If ^(n) 
denotes the product of the integers <n and prime to n, he proved by induc- 
tion that ^(p')= — 1 (mod p') for p an odd prune. For, if pi, . . ., p„ are 
the integers < p' and prime to it, then pi +jp', . . ., Pn +ip' (i = 0, 1 , . . . , p — 1) 
are the integers < p'"^^ and prime to it. He proved similarly by induction 
that 1/^(2') = + ! (mod 2') if 7r>2. Evidently iA(2) = l (mod 2), »//(4)=-l 
(mod 4). If m = a''b^ . . . and n = l^, where I is a new prime, then \p(m)=e 
(mod m), \l/{n) = r) (mod n) lead by the preceding method to \l/{mn) = e'^^"^ 
(mod m), viz., 1, unless n = 2. The theorem now follows easily. 
E. Lucas^^^ noted that, if x is prime to n = AB . . ., where A, B,. . . are 
powers of distinct primes, and if is the 1. c. m. of (f){A), (l>{B),. . ., then 
x'^= 1 (mod n). In case A = 2^', A:> 2, we may replace (j){A) by its half. To 
get a congruence holding whether or not x is prime to n, multiply the former 
congruence by x", where a is the greatest exponent of the prime factors of n. 
Note that <}>-\-<T<n [Bachmann^^^' "^]. CarmichaeP^^ wrote X(n) for 0. 
E. Lucas^^^ found A^~^x^~^ in two ways by the theory of differences. 
Equating the two results, we have 
(p-i)!=(p-ir^-(^7^)(p-2ri+...-(^:^)i 
Each power on the right is = 1 (mod p) . Thus 
(p-l)!=(l-l)p-i-l=-l (modp). 
P. A. MacMahon^^^ proved Fermat's theorem by showing that the 
number of circular permutations of p distinct things n at a time, repetitions 
allowed, is 
h<l>(d)p^^', 
ft 
where d ranges over the divisors of n. For n a prime, this gives 
p"+(n — l)p=0, p"=p (mod n). ^^ 
Another specialization led to Euler's generalization. 
E. Maillet^^^ applied Sylow's theorem on subgroups whose order is 
the highest power p'' of a prime p dividing the order m of a group, viz., 
"^British Association Report, 1888, 580-1. 
""Lineaire Congruenties, Diss. Amsterdam, 1890, 104-114. 
""BuU. Ac. Sc. St. P6tersbourg, 33, 1890, 496. 
">Mathesis, (2), 1, 1891, 11; Th^orie des nombres, 1891, 432. 
""Proc. London Math. Soc, 23, 1891-2, 305-313. 
"'Recherches sur les substitutions, Th§se, Paris, 1892, 115. 
