70 
History of the Theory of Numbers. 
[Chap. HI 
A. L. Crelle^^ proved the generalized Wilson theorem. By pairing each 
root <T of x-=l (mod s) with the root s—a, and each integer a<s, prime to 
s and not a root, with its associated number a', where aa'=\ (mod s), we 
see that the product of all the integers <s and prime to s is = + 1 or —1 
(mod s) according as the number n of pairs of roots o-, s—o- is even or odd. 
To find n, express s in every way as a product of two factors u, v, whose 
g. c. d. is 1 or 2; in the respective cases, each factor pair gives a single root 
(T or two roots. Treating four subcases at length it is shown that the num- 
ber of factor pairs is 2^" in each case, where k is the number of distinct odd 
primes dividing s ; and then that n is odd if s = 4, p"* or 2p", but even if n 
is not of one of these three forms. 
A. Cauchy^^" proved Fermat's theorem as had Leibniz.'* 
V^^ (S. Earnshaw?) proved Wilson's theorem by Lagrange's method and 
noted that, if Sr is the sum of the products of the roots of AqX"'+Aix"'~^-\- . . . 
= (mod p) taken r at a time, then AoSi — { — iyAi is divisible by p. 
Paolo Gorini^" proved Euler's theorem 6'=1 (mod A), where t=(f>(A), 
by arranging in order of magnitude the integers (A) p', p", . . . , p^'^ which 
are less than A and prime to A. After omitting the numbers in (A) which 
are di\'isible by h, we obtain a set (B) q',. . ., q^^\ Let 5^"^ be the least of 
the latter which when increased by A gives a multiple of h : 
(C) 
g(-)+A = p^'^^6. 
V 
(a-1) 
i(0 
The numbers* (A) coincide with those in sets (B) and (D) : 
(D) p%p"b,...,p^-%. 
Hence by multiplication and cancellation of p', 
(F) q'...qH''-^ = p^''\..p^ 
To each number (B) add the least multiple of A which gives a sum divisible 
by b, say (G) q'+g'A,..., q^^+g^^A. The least of these is q^''^-\-A = 
p^^^h, by (C). Each number (G) is <6A and all are distinct. The quo- 
tients obtained by di\'iding the numbers (G) by h are prime to A and hence 
included among the p^^V-j P^'\ whose number is t—a-\-l=l, so that 
each arises as a quotient. Hence 
(H) n(g«+^«A)=PA+g'. . .g^'^ = p' 
Combine this with (F) to eliminate the p's. 
pW 
-a+l 
q'. . .g«6''-W-"+i = PA+5'. . .q 
(0 
■Q 
We get 
6'-l = QA. 
"Jour, fur Math., 20, 1840, 29-56. Abstract in Bericht Akad. Wiss. Berlin, 1839, 133-5. 
»8aM6m. Ac. Sc. Paris, 17, 1840, 436; Oeuvres, (1), 3, 163-4. 
"Cambr. Math. Jour., 2, 1841, 79-81. 
"Annali di Fisica, Chimica e Mat. (ed., G. A. Majocchi), Milano, 1, 1841, 255-7. 
*To follow the author's steps, take A = 15, 6 = 2, whence « = 8, i = 4, (A) 1, 2, 4, 7, 8, 11, 13, 14; 
(B) 1, 7, 11, 13; (C) 1 + 15 = 8-2, ?(<>) =8, a = 5; (D) 2, 4, 8, 14; (F) 171113 2« = 8111314; 
(G) 1 + 15,7 + 15, 11+15;13 + 15, each g = l; the quotients of the latter by 2 are 8, 11, 13, 
14, viz., last four in (A); (H) P.15 + 1.7.11.13=8.11.13.14.2«; the second member ifl 
1-71113 2" by (F). Hence 171113 (28-l) = 15P. 
