Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 267 
E. Catalan^^ noted that 
e::r)'(?)"*e)-«" 
F. Gomes Teixeira^^ discussed the result due to Weill. ^® 
De Presle^® proved that 
{k-\-l)(k+2)...{k+hl) . ^ 
nw ^ ^^ ^^^'' 
being the product of an evident integer by {hl)\/{U{h\y}. , 
E. Catalan^^ noted that, if n is prime to 6, 
(2n-4)! 
nl{n-2)\ 
H. W. Lloyd Tanner^^ proved that 
= integer. 
= integer. 
{\,\...\,\ng\y 
L. Gegenbauer stated and J. A. Gmeiner^^ proved arithmetically that, 
if n=Sjrja_,iaj2. • Oys, the product 
m{m+k)(m-h2k) . . . {m+{n-l)k}k''-' 
is divisible by 
where m, k, n, an,- ■ ■, o,rs are positive integers. This gives Hermite's'^' 
result by taking r = s = l. The case m = A; = l,s = 2, is included in the result 
by Weill.26 
Heine^^" and A. Thue^° proved that a fraction, whose denominator is k\ 
and whose numerator is a product of k consecutive terms of an arithmetical 
progression, can always be reduced until the new denominator contains only 
such primes as divide the difference of the progression [a part of Her- 
mite's^^ result]. 
F. Rogel'*^ noted that, if P be the product of the primes between (p — 1)/2 
and p + 1, while n is any integer not divisible by the prime p, 
(n-l)(n-2). ..{n-p-^l)P/p=0 (mod P). 
S. Pincherle^^ noted that, if n is a prime, 
P={x+l){x+2) . ..(x+n-l) 
is divisible by n and, if x is not divisible by n, by n !. If n = Up", P is divisible 
="Nouv. Ann. Math., (3), 4, 1885, 487. Proof by Landau, (4), 1, 1901, 282. 
35Archiv Math. Phys., (2), 2 1885, 265-8. ^eBuU. Soc. Math. France, 16, 1887-8, 159. 
"M6m. Soc. Roy. Sc. Li^ge, (2), 15, 1888, 111 (Melanges Math. III). Mathesis, 9, 1889, 170. 
"Proc. London Math. Soc, 20, 1888-9, 287. ^QMonatshefte Math. Phys., 1, 1890, 159-162. 
"a Jour, fur Math., 45, 1853, 287-8. Cf. Math. Quest. Educ. Times, 56, 1892, 62-63. 
"Archiv for Math, og Natur., Kristiania, 14, 1890, 247-250. 
"Archiv Math. Phys., (2), 10, 1891, 93. 
"Rendiconto Sess. Accad. Sc. Istituto di Bologna, 1892-3, 17. 
