74 HISTORY OF THE THEOKY OP NUMBERS. [CHAP, in J. Toeplitz83 gave Lagrange's proof of Wilson's theorem. M. A. Stern84 made use of the series for log (1— x) to show that . . . 1— # Multiply together the series for ex, e*/2, etc. By the coefficient of xp, Take p a prime. No term of s has a factor p in the denominator. Hence P V. A. Lebesgue85 obtained Wilson's theorem by taking x=p — I in If P is a composite number ?*4, (P— 1)! is divisible by P. He (p. 74) attributed Ivory's33 proof of Fermat's theorem to Gauss, without reference. G. L. Dirichlet86 gave Horner's37 and Euler's14 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 somewhat simpler than the analogous proofs. Jean Plana87 used the method of powering. Let N~pkpil .... For M prime to N, Mp~l = 1 +pQ. Hence Thus for e — (p(pkpiki)j Me—l is divisible by pk and p±kl 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 — 1 , subtract the expansion of (1 — l)p~l and write the resulting series in reverse order: H. F. Talbot88 gave Euler's12 proof of Fermat's theorem. J. Blissard88a proved the last statement of Euler.9 C. Sardi89 gave Lagrange's proof of Wilson's theorem. P. A. Fontebasso90 proved (2) for x = a by finding the first term of the ath order of differences of ya, (y-{-hy, (y+2h)a, ... and then setting y = 0, h = 1. 83Archiv Math. Phys., 32, 1859, 104. "Lehrbuch der Algebraischen Analysis, Leipzig, 1860, 391. "Introd. th^orie des nombres, Paris, 1862, 80, 17. MZalilentheorie (ed. Dedekind), §§19, 20, 127, 1863; ed. 2, 1871; ed. 3, 1879, ed. 4, 1894. 87Mem. Acad. Turin, (2), 20, 1863, 148-150. "Trans. Roy. Soc. Edinburgh, 23, 1864, 45-52. BWMath. Quest. Educ. Times, 6, 1866, 26-7. "Giornale di Mat., 5, 1867, 371-6. BOSaggio di una introd. arit. trascendente, Treviso, 1867, 77-81.