74 History of the Theory of Numbers. [Chap, hi 
J. Toeplitz^ gave Lagrange's proof of Wilson's theorem. 
M. A. Stern^ made use of the series for log (1 — x) to show that 
1+x+xH. . . =-^ = e'+*'/2+^/3+...^ 
l—x 
Multiply together the series for e', e'*^^, etc. By the coeflBcient of x^. 
p! P' (p-2)!' ••• 
Take p a prime. No term of s has a factor p in the denominator. Hence 
(1-s) • (p-l)! = ^-tfcli^ = integer. 
P 
V. A. Lebesgue^^ obtained Wilson's theorem by taking x = p — l in 
p X Hk+l) . . . {k-\-p-2) =x(x+l) . . . (x+p-1). 
k=i 
If P is a composite number ?^4, (P — 1)! is di\'isible by P. He (p. 74) 
attributed Ivory's^^ proof of Fermat's theorem to Gauss, without reference. 
G. L. Dirichlet^^ gave Horner's" and Euler's^^ proof of Euler's theorem 
and derived it from Fermat's by the method of powering. His proof (§38) 
of the generalized Wilson theorem is by associated numbers, but is some- 
what simpler than the analogous proofs. 
Jean Plana" used the method of powering. Let N = p^pi' .... For M 
prime to N, M^~'^ = 1 +pQ. Hence 
Thus for e = (p{p^pi'), M" — ! is divisible bj' p'' and p/' and hence by their 
product, etc. Plana gave also a modification of Lagrange's proof of Wilson's 
theorem by use of (2) ; take x=a = p — l, subtract the expansion of (1 — 1)""^ 
and write the resulting series in reverse order: 
(p-l)!+l = (^2^)(2^-^-l)-(V)(3^-^-l)+... 
-(^:D](p-2)^-^-lt + 1(p-i)''-'-if- 
H. F. Talbot^^ gave Euler's^^ proof of Fermat's theorem. 
J. Blissard^^" proved the last statement of Euler.^ 
C. Sardi^^ gave Lagrange's proof of Wilson's theorem. 
P. A. Fontebasso^*^ proved (2) for x = a by finding the first term of the 
ath order of differences ofy'',{y+hy,{y-\-2hy,. . . and then setting y = 0,h = l. 
^'Archiv Math. Phys., 32, 1859, 104. 
"Lehrbuch der Algebraischen Analysis, Leipzig, 1860, 391. 
"Introd. thdorie des nombres, Paris, 1862, 80, 17. 
"Zahlentheorie (ed. Dedekind), §§19, 20, 127, 1863; ed. 2, 1871; ed. 3, 1879, ed. 4, 1894. 
8'Mem. Acad. Turin, (2), 20, 1863, 148-150. 
"Trans. Roy. Soc. Edinburgh, 23, 1864, 45-52. 
ss^'Math. Quest. Educ. Times, 6, 1866, 26-7. 
"Giomale di Mat., 5, 1867, 371-6. 
•"Saggio di una introd. arit. trascendente, Treviso, 1867, 77-81. 
