Chap. Ill] FeRMAT's AND WiLSON's THEOREMS. 71 
E. Lionnet^^ proved that, if p is an odd prime, the sum of the mth powers 
of 1,. . ., p — 1 is divisible by p for 0<m<p — l. Hence the sum P^ of 
the products of 1, ... ,p — 1 taken w at a time is divisible by p [Lagrange^^]. 
Since 
(l + l)(l+2)...(H-p-l) = l+Pi+P2+...H-Pp-2+(p-l)!, 
l + (p — 1)! is divisible by p. 
E. Catalan^^ gave the proofs by Ivory^^ and Horner.^^ C. F. Arndt^^ gave 
Horner's proof; and proved the generalized Wilson theorem by associated 
numbers. O. Terquem^^ gave the proofs by Gauss^^ and Dirichlet.^° 
A. L. Crelle^^ republished his proof'*'' of Wilson's theorem, as well as 
that by Gauss^° and Dirichlet.^° Crelle^^ gave two proofs of the generalized 
Wilson theorem, essentially that by Minding^^ and that given by himself.^^ 
If fj, is the number of distinct odd prime factors of z, and 2^" is the highest 
power of 2 dividing z, and r is a quadratic residue of z, then (p. 150) the 
number n of pairs of roots ±x of x^=r (mod z) is 2""^ if m = or 1, 2" if 
m = 2, 2"'^^ if w>2. Using the fact (p. 122) that the quadratic residues of 
z are the e=(f){z)/(2n) roots of r*=l (mod z), it is shown (p. 173) that, if v 
is any integer prime to z, y*'^^^'''*=l (mod z), "a, perfection of the Euler- 
Fermat theorem." 
L. Poinsot^^ failed in his attempt to prove the generalized Wilson 
theorem. He began as had Crelle.^^ But he stated incorrectly that the 
number n of pairs of roots =^x of x^^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 = y = 4; but for the root x = 7 (or for x = 17), a: ± 1 yield 
P, Q = 3, 8, and 6, 4. To correct another error by Poinsot, let n be the number 
of distinct odd prime factors of s and let 2"* be the highest power of 2 dividing 
s; then y = 2''-^ 2", 3-2''-^ or 2"+^ according as w = 0, 1, 2, or ^3, whereas 
[Crelle*^^] n = 2''-\ 2''-\ 2^ 2"+^ No difficulty is met (pp. 53-5) in case the 
modulus is a power of a prime. He noted (p. 33) that if Vi, r2, . . . are the 
integers <N and prime to N, and tt is their product, they are congruent 
modulo N to tt/ti, Tr/ra, ..., whence T=Tr''~^ (mod N), where v=(f){N). 
Thus, by Euler's theorem, 7r^= 1. This does not imply that 7r= =*= 1 as cited 
by Aubry,!" pp. 30O-I. 
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 xth ver- 
tex following it, the new vertex with the a:th 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. 
«/6td., 462-4. 
«Archiv Math. Phys., 2, 1842, 7, 22, 23. 
"Nouv. Ann. Math., 2, 1843, 193; 4, 1845, 379. 
«Jour. fur Math., 28, 1844, 176-8. 
«8/bid., 29, 1845, 103-176. 
«^Jour. de Math., 10, 1845, 25-30. German exposition by J. A. Grunert, Archiv Math. Phys., 
7, 1846, 168, 367. 
