76 History of the Theory of Numbers. [Chap, hi 
is zero ii n<m, but, if ti^tt^, equals 
where 
For n = m, the initial sum equals Em = rn\. 
P. Mansion^® noted that Euler's theorem may be identified with a 
property of periodic fractions [cf. Laisant^^]. Let N be prime to R. Taking 
R as the base of a scale of notation, divide 100. . .by A^ and let gi . . .g„ be 
the repetend. Then (72" — l)/iV = Q'i. . .5„. Unless the n remainders r^ 
exhaust the integers <N and prime to A^, we divide r/ 00. . .by A^, where 
r/ 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 R'^'-^ — l. [At 
bottom this is Euler's^* proof.] 
P. Mansion^^ reproduced this proof, made historical remarks on the 
theorem and indicated an error by Poinsot.^^ 
Franz Jorcke^^ reproduced Euler's^^ proof of Wilson's theorem. 
G. L. P. V. Schaewen^^ proved (2) with a changed to —p, by expanding 
the binomials. 
Chr. ZeUer^o" proved that, for n ?^ 4, 
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. Cayley^°^ proved Wilson's theorem as had Petersen.^^ 
E. Schering^^^ took a prime to m = 2'pi''\ . .p'", where the p's are dis- 
tinct odd primes and proved that x^=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 \p{m)=2'', 2""^^ or 
2"''"^, according as 7r<2, 7r = 2, or 7r>2. Let a be a fixed quadratic residue 
of m and denote the roots by ^aj (j = l,. . ., 4'/2). Set a.- =m—aj. The 
<}>{m)—\f/{ni) integers <m and prime to m, other than the ay, a/, may be 
denoted by aj, a/ (i = |iA+lj- • •? 20)j where aja'j=a (mod m). From the 
latter and —aja/=a (i = l, . . ., ^/2), we obtain, by multiplication, 
^i^(m) = (_j)}*(m)^^ .r^ (mod w), 
••Messenger Math., 5, 1876, 33 (140); Xouv. Corresp. Math., 4, 1878, 72-6. 
"Th6orie des nombres, 1878, Gand (tract). 
•*Uber Zahlenkongruenzen, Progr. Fraustadt, 1878, p. 31. 
"Die Binomial Coefficienten, Progr. Saarbriicken, 1881, p. 20. 
looBull. des sc. math, astr., (2), 5, 1881, 211^. 
"'Messenger of Math., 12, 1882-3, 41; CoU. Math. Papers, 12, p. 45. 
iwActa Math., 1, 1882, 153-170; Werke, 2, 1909, 69-86. 
