CHAP. Ill] FEKMAT'S AND WILSON'S THEOREMS. 81 P. Bachmann129 proved the first statement of Lucas.110 He gave as a "new" proof of Euler's theorem (p. 320) the proof by Euler,14 and of the generalized Wilson theorem (p. 336) essentially the proof by Arndt.70 J. W. Nicholson130 proved the last formula of Grunert.36 Bricard131 changed the wording of Petersen's94 proof of Fermat's theorem. Of the qp numbers with p digits written to the base q, omit the q numbers with a single repeated digit. The remaining <f— q numbers fall into sets each of p distinct numbers which are derived from one another by cyclic permutations of the digits. G. A. Miller132 proved the generalized Wilson theorem by group theory. The integers relatively prime to g taken modulo g form under multiplication an abelian group of order <fr(g) which is the group of isomorphisms of a cyclic group of order g. But in an abelian group the product of all the elements 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 pa, 2pa or 4 if its group of isomorphisms contains a single element of period 2. V. d'Escamard133 reproduced Sylvester's78 proof of Wilson's theorem. K. Petr134 gave Petersen's94 proof of Wilson's theorem. Prompt135 gave an obscure proof that 2P~1— • 1 is divisible by the prime p. G. Arnoux136 proved Euler's theorem. Let X be any one of the t>=0(7n) integers a, )8, 7, . . ., prime to m and <m. We can solve the congruences aa'=/?/3'=7y' = . . . =X (mod m). Here a', 0', . . .form a permutation of a, j8, ____ Thus aa7^.. = (a/3...)2E=\«. In particular, for X = l, we get (a/3. . .)2=1. Hence for any X prime to m, X'=l (mod m). [CL Dirichlet,40 Schering,102 C. Moreau.123] R. A. Harris136a proved that (a/3 . . .)2 = 1 as did Arnoux136, but inferred falsely that a. 0. . .35=1=1. A. Aubry137 started, as had Waring in 1782, with ^^Yn+AYn.l+ . . . +MY2+Y1} where Yt = x(x-i). . .(x-p+1). Then Summing for $-1,. . ., p — 1 and setting SA.= l*+2*+. . .+(p — l)k, we get ""Niedere Zahlentheorie, I, 1902, 157-8. 13QAmer. Math. Monthly, 9, 1902, 187, 211. 131Nouv. Ann. Math., (4), 3, 1903, 340-2. 132Annals of Math., (2), 4, 1903, 188-190. Cf. V. d'Escamard, Giornale di Mat., 41, 1903, 203-4; U. Scarpis, ibid., 43, 1905, 323-8. "'Giornale di Mat., 43, 1905, 379-380. l3<Casopis, Prag, 34, 1905, 164. l3BRemarques BUT le th<k>reme de Fermat, Grenoble, 1905, 32 pp. "•Arithme'tique Graphique; Fonctions Arith., 1906, 24. 138flMath. Magazine, 2, 1904, 272. ll7L'euseignement math., 9, 1907, 434-5, 440.