CHAP, ill] FERMAT'S AND WILSON'S THEOREMS. 77 where the ry are the integers < m and prime to m. Taking a = 1, we have the generalized Wilson theorem. Applying a like argument when a is a quadratic non-residue of m [Minding48], we get This investigation is a generalization of that by Dirichlet.40 E. Lucas103 wrote Xp for x(x+l). ..(x+p-1), and TJ for the sum of the products of 1,..., p taken q at a time. Thus Replacing p by 1,..., n in turn and solving, we get where pl p2 jtn-p-H 1 T1* pn—p the subscript p — 1 on the F's being dropped. After repeating the argument by Tchebychef75, Lucas noted that, if p is an odd prime, An»p+1«l or 0 (mod p), according as p — 1 is or is not a divisor of n. G. Wertheim104 gave Dirichlet's86 proof of the generalized Wilson theorem; also the first step in the proof by Arndt.70 W. E. Heal105 gave without reference Euler's14 proof. E. Catalan106 noted that if 2n+l is composite, but not the square of a prime, n! is divisible by 2n+l; if 2n+l is the square of a prime, (n!)2 in divisible by 2n+l. C. Garibaldi107 proved Fermat's theorem by considering the number N of combinations of ap elements p at a time, a single element being selected from each row of the table epl ep2.. .epa. From all possible combinations are to be omitted those containing demonta from exactly n rows, for n=l, . . ., p — 1. Let An denote the number of combinations p at a time of an elements forming n rows, nuch that, in eueh combination occur elements from each row. Then 1MBull. Soc. Math. France, 11, 1882-3, (50-71; Mutli(wi«, ,'i, l«K,'i, 25-H. 1MElemente der Zahlcnthcorie, 1887, 180-7; AnfunKHgrthuh^ dor Zahlcnlchn-, 11)02 a-iil .r> (331-2). 105Annals of Math., 3, 1887, 97-98. 1MM6m. soc. roy. sc. Li6gc, (2), 15, 1888 (Melanges Math., Ill, 1887 139) 107Giornale di Mat., 26, 1888, 197.