CHAP. Hi] FERMAT'S AND WILSON'S THEOBEMS. 71 E. Lionnet61 proved that, if p is an odd prime, the sum of the mth powers of 1,. ."., p — 1 is divisible by p forO<m<p — 1. Hence the sum Pm of the products of 1, . . . ,p — 1 taken m at a time is divisible by p [Lagrange18]. Since 1 + (p — 1) ! is divisible by p. E. Catalan62 gave the proofs by Ivory33 and Homer.37 C. F. Arndt63 gave Horner's proof; and proved the generalized Wilson theorem by associated numbers. 0. Terquem64 gave the proofs by Gauss28 and Dirichlet.40 A. L. Crelle66 republished his proof47 of Wilson's theorem, as well as that by Gauss30 and Dirichlet.40 Crelle66 gave two proofs of the generalized Wilson theorem, essentially that by Minding48 and that given by himself.58 If jtt is the number of distinct odd prime factors of 2,, and 2m is the highest power of 2 dividing z, and r is a quadratic residue of 2, then (p. 150) the number n of pairs of roots =*=£ of ofer (mod z) is 2"'1 if w=0 or 1, 2M if w=2, 2"+1 if m>2. Using the fact (p. 122) that the quadratic residues of z are the e=<j>(z)/(2ri) roots of r€=l (mod 2), it is shown (p. 173) that, if v is any integer prime to z, v*(*)/ns=i (mod 2), "a perfection of the Euler-Fermat theorem." L. Poinsot67 failed in his attempt to prove the generalized Wilson theorem. He began as had Crelle.68 But he stated incorrectly that the number n of pairs of roots =*=a; of x2=l (mod s) equals the number v of ways of expressing s as a product of two factors P, Q whose g. c. d. is 1 or 2. For each pair =*= x, it is implied that x-^1 and x+1 uniquely determine P, Q. For s = 24, n = v = 4; but for the root x = 7 (or for x = 17), x =*= 1 yield P, Q = 3, 8, and 6, 4. To correct another error by Poinsot, let /* be the number of distinct odd prime factors of s and let 2m be the highest power of 2 dividing s; then 0=2*~1, 2M, SW"1 or 2M+1, according as w = 0, 1, 2, or ^3, whereas [Crelle66] n = 2"~l, 2*"1, 2M, 2"+1. No difficulty is met (pp. 53-5) in case the modulus is a power of a prime. He noted (p. 33) that if rx, r2, . . . are the integers <N and prime to Nt and TT is their product, they are congruent modulo N to T/n, ir/r2,.. ., whence ir^n""1 (mod AT), where v = (j)(N). Thus, by Euler's theorem, w2= 1. This does not imply that TT= =*= 1 as cited by Aubry,137 pp. 300-1. Poinsot (p. 51) proved Euler's theorem by considering a regular polygon of N sides. Let x be prime to N and < N. Join any vertex with the zth vertex following it, the new vertex with the icth vertex following it, etc., thus defining a regular (star) polygon of N sides. With the same x, derive "Nouv. Ann. Math., 1, 1842, 175-6. **Ibid., 462-4. "Archiv Math. Phys., 2, 1842, 7, 22, 23. MNouv. Ann. Math., 2, 1843, 193; 4, 1845, 379. "Jour, fiir Math., 28, 1844, 176-8. w/6id., 29, 1845, 103-176. "7Jour. de Math., 10, 1845, 25-30. German exposition by J. A. Grunert, Archiv Math. Phys., 7, 1846, 168, 367.