Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 265 
is an integer, and wrote L^ for its residue modulo p^+\ Set 
:.= P 
He proved that tt^ is divisible by p*, where s = Hi+Sj;io2^. If t„ is the 
first one of the numbers tq, ti, . . . which is <p — 1, tt^ is divisible by p", 
k 
A. Cunningham^^ proved that if / is the highest power of the prime z 
dividing p, the number of times p is a factor of p"! is the least of the 
numbers ^„ 2^«-^+i-l 
for the various primes z dividing p. 
W. Janichen^^ stated and G. Szego proved that 
i:tx{n/d)v{d)=ci>{n)/{v-l), 
summed for the divisors d of n, where v{d) is the exponent of the highest 
power of p (a prime factor of n) which divides d\, for /x as in Ch. XIX. 
Integral Quotients Involving Factorials. 
Th. Schonemann^^ proved, by use of symmetric functions of pth roots 
of unity, that if b is the g. c. d. of fjL,v,'..., 
8-{m-l)l 
.ii/l 
= integer, {m—fx+v-\- . . .). 
fJLlVl 
He gave (p. 289) an arithmetical proof by showing that the fractions 
obtained by replacing 8 by fi, v, .. . are integers. 
A. Cauchy^^ proved the last theorem and that 
^ -^ ^ = integer, {m = a-\- . . .+k). 
a\. . .k\ 
D. Andre^*^ noted that, except when n = l, a = 4, n(n + l). . .(na — 1) is 
not or is divisible by a" according as a is a prime or not. 
E. Catalan^^ found by use of elliptic functions that 
{m-\-n-\)\ (2m) ! (2n) ! 
m!n! m\n\{m-\-n)\ 
are integers, provided m, n are relatively prime in the first fraction. 
^^L'intermediaire des math., 19, 1912, 283-5. Text modified at suggestion of E. Maillet. 
"Archiv Math. Phys., (3), 13, 1908, 361; 24, 1916, 86-7. 
»8Jour. fur Math., 19, 1839, 231-243. 
"Comptes Rendus Paris, 12, 1841, 705-7; Oeuvres, (1), 6, 109. 
20N0UV. Ann. Math., (2), 11, 1872, 314. 
"/fttd., (2), 13, 1874, 207, 253. Arith. proofs, Amer. Math. Monthly, 18, 1911, 41-3. 
