76 HlSTOKY OP THE THEORY OF NUMBERS. [CHAP. Ill is zero if ?i<m, but, if n^m, equals where (2) (&-2)n- • • •+(-Dt For n = ra, the initial sum equals Em = m !. P. Mansion96 noted that Euler's theorem may be identified with a property of periodic fractions [cf . Laisant91] . Let N be prime to R. Taking R as the base of a scale of notation, divide 100. . .by N and let gx . . .qn be the repetend. Then (Rn— l)/N=q1. . .qn. Unless the n remainders r,-exhaust the integers <N and prime to N, we divide r/ 00. . .by N, where 7*1' is one of the integers distinct from the r^ and obtain n new remainders r/. In this way it is seen that n divides <p(N) , so that N divides Rv(N) — 1 . [At bottom this is Euler's14 proof.] P. Mansion97 reproduced this proof, made historical remarks on the theorem and indicated an error by Poinsot.67 Franz Jorcke98 reproduced Euler's22 proof of Wilson's theorem. G. L. P. v. Schaewen" proved (2) with a changed to — p, by expanding the binomials. Chr. Zeller100 proved that, for n^4, (n-3)*+ . . . is divisible by n unless n is a prime such that n—1 divides x, in which case the expression is = — 1 (mod n). A. Cayley101 proved Wilson's theorem as had Petersen.94 E. Schering102 took a prime to m = 2*p1in. . .p/M, where the p's are distinct odd primes and proved that x2=a (mod m) has roots if and only if a is a quadratic residue of each Pi and if a=l (mod 4) when 7r = 2, a=l (mod 8) when 7r>2, and then has \l/(m) roots, where ^(m)==2M, 2M+1 or 2M+2, according as ?r<2, 7r = 2, or 7r>2. Let a be a fixed quadratic residue of m and denote the roots by =ta,- (j = l, . . ., t///2). Set a/ = m— a;. The <t>(m)—\]/(m) integers <m and prime to m, other than the a;, a/, may be denoted by a,-, a/ 0"=i^+l, . . ., J0), where aXy=a (mod m). From the latter and — aya/=a (j = 1, . . . , ^/2), we obtain, by multiplication, a^(»)s(-i)W'(m)ri> . >r^ (mod m)j ^Messenger Math., 5, 1876, 33 (140); Nouv. Corresp. Math., 4, 1878, 72-6. "ThSorie des nombres, 1878, Gand (tract). "Uber Zahlenkongrueazen, Progr. Fraustadt, 1878, p. 31. "Die Binomial Coefficienten, Progr. Saarbriicken, 1881, p. 20. l°°Bull. des sc. math, astr., (2), 5, 1881, 211-4. 101Messenger of Math., 12, 1882-3, 41; CoU. Math. Papers, 12, p. 45. lMActa Math., 1, 1882, 153-170; Werke, 2, 1909, 69-86.