88 
History of the Theory of Numbers. 
[Chap. Ill 
E. Malo^^ employed integers -4/ and set u = afz, 
Since f^ d='Eu''/k {k = n, m+n, 2m-\-n, 
e= 
w 
l-u" 
=ScOpX''-Ma:. 
2^x^=2^^'^' 
P 
k 
p k 
p-ltkt 
where k takes the values n, w+n, . . .which are ^p/fi. If no prime factor 
of such a k occurs in the denominator of the expansion of cop/p, the latter 
is an integer; this is the case if p is a prime and /x= 2. For w = n = l, ix = 2, 
^(»-)-(2)-(3)- 
+ 
■ax 
a-3. 
.0-2 
we get o3p = a^—a and hence Fermat's theorem. 
L. Kronecker^^ generalized Fermat's and Wilson's theorems to modular 
systems. 
R. Le Vavasseur^^ obtained a result evidently equivalent to that by 
Moore^^^ for the non-homogeneous case Xm = l' 
M. Bauer^^^ proved that if n = p'm, where m is not divisible by the odd 
prime p, and Oi, . . ., a< are the t=<l>{n) integers <n and prime to n, 
{x-ai) . . .(a:-a,) = (xP-i-l)'/<P-'Hmod p'), 
identically in a:. If p = 2 and 7r> 1, the product is identically congruent to 
{x'^ — iy^^. 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 2*+ 1, the 
values are 
d 
2q 
4 
P 
2 
n 
2q 
4 
p-, 2p» 
2^2"5lg2... 
M. Bauer^*'' determined how n and N must be chosen so that x" — 1 
shall be congruent modulo A^ 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, a;" — ! 
is congruent modulo p° to a product of linear functions only when p=l 
(mod n), a arbitrary, or when n = p'm, a = l, p = l (mod m). 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.^^^ 
M. Bauer^^^ noted that, if n = 'p'm, where m is not divisible by the odd 
prime p, 
n(a:-i) = (xP-x)"/P (mod p'). 
t=i 
>8»L'interm6diaire des math., 7, 1900, 281, 312. 
^"Vorlesungen uber Zahlentheorie, I, 1901, 167, 192, 220-2. 
'"Comptes Rendus Paris, 135, 1902, 949; Mdm. Ac. Sc. Toulouse, (10), 3, 1903, 39-48. 
•"Nouv. Ann. Math., (4), 2, 1902, 256-264. 
"'Math. Nat. Berichte aus Ungarn, 20, 1902, 34-38; Math. 6s Phys. Lapok, 10, 1901, 274-8 
(pp. 145-152 relate to the "theory of Fermat's congruence"; no report is available). 
"8Amer. Jour. Math., 11, 1888; 13, 1891. 
'"Math. 68 Phys. Lapok, 12, 1903, 159-160. 
