Chap. Ill] FeRMAT's AND WiLSON's THEOREMS. 81 
P. Bachmann^^^ proved the first statement of Lucas."*' He gave as a 
"new" proof of Euler's theorem (p. 320) the proof by Euler,^^ and of the 
generaUzed Wilson theorem (p. 336) essentially the proof by Arndt.^° 
J. W. Nicholson^^^ proved the last formula of Grunert.^^ 
Bricard^^^ changed the wording of Petersen's^^ proof of Fermat's theorem. 
Of the q^ numbers with p digits written to the base q, omit the q numbers 
with a single repeated digit. The remaining q^—q numbers fall into sets 
each of p distinct numbers which are derived from one another by cyclic 
permutations of the digits. 
G. A. Miller^^^ proved the generalized Wilson theorem by group theory. 
The integers relatively prime to g taken modulo g form under multiplica- 
tion an abelian group of order (f){g) which is the group of isomorphisms of a 
cyclic group of order g. But in an abelian group the product of all the ele- 
ments is the identity if and only if there is a single element of period 2. 
It is shown that a cyclic group is of order p", 2p* or 4 if its group of isomor- 
phisms contains a single element of period 2. 
V. d'Escamard^^^ reproduced Sylvester's''^ proof of Wilson's theorem. 
K. Petr^^* gave Petersen's®^ proof of Wilson's theorem. 
Prompt^^® gave an obscure proof that 2^~^ — 1 is divisible by the prime p. 
G. Arnoux^^® proved Euler's theorem. Let X be any one of the 
v=4>{m) integers a, /3, 7, . . ., prime to m and <m. We can solve the con- 
gruences 
aa'=/3/3'=77'= . . . =\ (mod m). 
Here a', jS', . . .form a permutation of a, |S, . . . . Thus 
In particular, for X = l, we get (a/3. . .)^=1. Hence for any X prime to m, 
V=\ (mod m). [Of. Dirichlet/" Schering,i°2 C. Moreau.i^^] 
R. A. Harris^^^" proved that (aj8 . . .)^ = 1 as did Arnoux^^^, but inferred 
falsely that a./3 . . . = ± 1. 
A. Aubry^^'^ started, as had Waring in 1782, with 
where yp = a:(a; — 1). . .(x— p+1). Then 
x'»+i-a;'^= F„+i+A7„+ . . . -I-MF3+F2. 
Summing for x = 1, . . . , p — 1 and setting Sk = l*+2*+ . . . + (p — 1)*, we get 
_\n±l\_ \n\ , ,MJ3J , \2\ 
n+2 w+1 
«»Niedere Zahlentheorie, I, 1902, 157-8. ""Amer. Math. Monthly, 9, 1902, 187, 211. 
"iNouv. Ann. Math., (4), 3, 1903, 340-2. 
"2Annals of Math., (2), 4, 1903, 188-190. Cf. V. d'Escamard, Giornale di Mat., 41, 1903, 
203-4; U. Scarpis, ihid., 43, 1905, 323-8. 
»"Giomale di Mat., 43, 1905, 379-380. i=^Casopis, Prag, 34, 1905, 164. 
^"Remarques sur le theorSme de Fennat, Grenoble, 1905, 32 pp. 
"'Arithm^tique Graphique; Fonctions Arith., 1906, 24. 
i»««Math. Magazine, 2, 1904, 272. "'L'enseignement math., 9, 1907, 434-5, 440. 
