Chap. Ill] FeRMAt's AND WiLSON's THEOREMS. 77 
where the Vj are the integers <m and prime to m. Taking a=l, we have 
the generahzed Wilson theorem. Applying a like argument when a is a 
quadratic non-residue of m [Minding^^], we get 
^J^(m)^^^_ .r^=(_l)^'^('») (mod m). 
This investigation is a generaUzation of that by Dirichlet.*" 
E. Lucas^"^ wrote Xp for x{x-\-l) . . .{x-\-p — l), and F^ for the sum of 
the products of 1, . . . , p taken g at a time. Thus 
^ ~rA p-i^ ~r • • • ~rA p-i^ — Xp. 
Replacing p by 1, . . ., n in turn and solving, we get 
where 
(-l)"-^+'A„-p+i = 
•plp2 pn— p+1 
1 r^ Y'*~p 
o...iri 
the subscript p — 1 on the F's being dropped. After repeating the argument 
by Tchebychef^^, Lucas noted that, if p is an odd prime, A„_p+i=l or 
(mod p), according as p — 1 is or is not a divisor of n. 
G. Wertheim^°^ gave Dirichlet's^^ proof of the generalized Wilson 
theorem; also the first step in the proof by Arndt.^° 
W. E. HeaP''^ gave without reference Euler's^'* proof. 
E. Catalan^°^ 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!)^ is 
divisible by 2n+l. 
C. Garibaldi^"^ 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 
en ^12. . .eia 
^pl ^p2 • • • ^pa. 
From all possible combinations are to be omitted those containing elements 
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, such that in each 
combination occur elements from each row. Then 
--iV-tO^- 
"'BuU. Soc. Math. France, 11, 1882-3, 69-71; Mathesis, 3, 1883, 25-8. 
i^Elemente der Zahlentheorie, 1887, 186-7; Anfangsgriinde der Zahlenlehre, 1902, 343-5 
(331-2). 
"'Annals of Math., 3, 1887, 97-98. 
»««M6m. soc. roy. so. LiSge, (2), 15, 1888 (Melanges Math., Ill, 1887, 139). 
"^Giornale di Mat., 26, 1888, 197. 
