80 History of the Theory of Numbers. [Chap, hi 
unity to each element and replacing p+1 by 1. Let m be the least number 
of repetitions of this process which will yield the initial permutation. For 
p a prime, m = l or p. There are p — 1 cases in which m = \. Hence 
(p — 1) ! — (p — 1) is divisible by p. Cf . Petersen.^^ 
Many proofs of (3), p. 63, have been given. ^^° 
D. von Sterneck^-^ gave Legendre's proof of Wilson's theorem. 
L. E. Dickson^-^ noted that, if p is a prime, p(p — 1) of the p! substitu- 
tions on p letters have a linear representation x'=ax-\-h, a^O (mod p), 
while the remaining ones are represented analytically by functions of degree 
> 1 which fall into sets of p^(p — 1) each, viz., aJ{x-^h)-\-c, where a is prime 
top. Hencep!—p(p — l) is a multiple of p^(p — l), and therefore (p — 1)!+1 
is a multiple of p. 
C. Moreau^-^ gave without references Schering's^°^ extension to any 
modulus of Dirichlet's^° proof of the theorems of Fermat and Wilson. 
H. Weber^'^ deduced Euler's theorem from the fact that the integers 
Km and prime to m form a group under multiplication, whence every 
integer belongs to an exponent dividing the order 0(m) of the group. 
E. Cahen^-^ proved that the elementary symmetric functions of 1,. . ., 
p — 1 of order <p — 1 are divisible by the prime p. Hence 
(a:-l)(a:-2). . .(a:-p+l)=xP-^+(p-l)! (modp), 
identically in x. The case x = l gives Wilson's theorem, so that also Fer- 
mat's theorem follows. 
J. Perott^-^ gave Petersen's^^ proof of Fermat's theorem, using q^ ''con- 
figurations" obtained by placing the numbers 1, 2,..., q into p cases, 
arranged in a line. It is noted that the proof is not vaUd for p composite; 
for example, if p = 4, g = 2, the set of configurations derived from 1212 by 
cyclic permutations contains but one additional configuration 2121. 
L. Kronecker^^^ proved the generalized Wilson theorem essentially as 
had Brennecke.^^ 
G. Candido^^^ made use of the identity 
aP+6P= (a+6)P-pa6(a+6)P-2+ . . . 
^^_^). p(p-2r+10...(p-r-l) .,,_^ ^^^ 
1-2. . .r 
Take p a prime and 6= —1. Thus a^ — a=(a — 1)^— (a — 1) (mod p). 
"«L'mterm6diaire des math., 3, 1896, 2&-28, 229-231; 7, 1900, 22-30; 8, 1901, 164. A. Capelli 
Giornale di Mat., 31, 1893, 310. S. Pincherle, ibid., 40, 1902, 180-3. 
"iMonatshefte Math. Phys., 7, 1896, 145. 
»»Annals of Math., (1), 11, 1896-7, 120. 
i"Nouv. Ann. Math., (3), 17, 1898, 296-302. 
i^Lehrbuch der Algebra, II, 1896, 55; ed. 2, 1899, 61. 
^"£l6ment3 de la throne des nombres, 1900, 111-2. 
>»'Bull. des Sc. Math., 24, I, 1900, 175. 
i"Vorlesungen uber Zahlentheorie, 1901, I, 127-130. 
"'Giomale di Mat., 40, 1902, 223. 
