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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*        pnp
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, Anp+1l 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, lK,'i, 25-H.
1MElemente der Zahlcnthcorie,   1887,   180-7;  AnfunKHgrthuh^ dor  Zahlcnlchn-,   11)02    a-iil .r>
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.