Chap. IX] DiVISIBILTY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 277 
...p\ P' = p'+^..p^-' with t = {v-l)/2, when p = 4A;-l; but P=pp''-^ 
pV-^ . ., P' = pY-'' pV"". • ., when p = 4A; + l. 
G. 01tramare^^° gave several algebraic series for the reciprocal of the 
binomial coefficient C^) and concluded that, if the moduli are primes, 
!+(-')= -2{(i)%(||)%(i|^J+ . . .} (mod 4^+1), 
2=+(-') = -K(iy+(riT+(wy+ ■ • •} ^^'^ *-+3)- 
V. Bouniakowsky^^^ considered the integers qi,. . ., Qs, each <N and 
prime to N, arranged in ascending order of magnitude. If X is any chosen 
integer ^s, multiply 
q, = N-qi, qs-i = N-q2,..., g,_x+i = iV-gx 
together and multiply the resulting equation by qi. . . q^^x- Apply the 
generalized Wilson theorem qi. . .g^+( — 1)^=0 (mod A'"). Hence 
9i?2- • •5x-gi?2. . .g.-x-f(-l)'+^=0 (mod N). 
For N a prime, we have s = N—l and 
X!(iV-l-X)!+(-l)'=0 (mod N) (l^XSN-l). 
C. A. Laisant and E. Beaujeux^^^ gave the last result and 
{'-.'} 
(-If (mod p), ^ = ^- 
F. G. Teixeira^^^ proved that if a=-2^''-^p-a, a<2p-l, 
a{a+l) . . .{a+2p-l)=3^-5\ . .{2p-iyp 
(mod a+a+l+a+2+...+a+2p-l). 
Thus, for p = 3, a = 1 , a = 95, 
95-96-97-98-99-100=32-52-3 (mod 585 = 95+ •• .+100). 
M. Vecchi^^^ noted that the final formula by Bouniakowsky^^^ follows 
by induction. Taking X = (iV— 1)/2, we get Lagrange's formula (4). 
From the latter, we get 
{3.5-7. . . (22/-l)}2| (^^^^) \f/2'^^{-l)'^ (mod p). 
The case y={p — l)/2 gives Arndt's"^ result 
(6) {3-5-7...(p-2)P=(-l)~ (modp). 
Vecchi^^^ proved that, if v is the number of odd quadratic non-residues 
of a prime p = 4n+3, then 1-3-5. . .(p — 2) = ( — 1)" (mod p). If n is the 
number of non-residues <p/2, 1-3-5. . .{p-2)={-iy+^2^''-^^^^ (mod p). 
""M^m. de I'lnstitut Nat. Genevois, 4, 1856, "sjomal de Sciencias Math, e Astr., 3, 1881, 
33-6. 105-115. 
"iBull. Ac. Sc. St. P^tersbourg, 15, 1857, 202-5. i^^Periodico di Mat., 16, 1901, 22-4. 
"2Nouv. Corresp. Math., 5, 1879, 156 (177). '^Hbid., 22, 1907, 285-8. 
