266 History of the Theory of Numbers. [Chap, ix 
P. Bachmann^- gave arithmetical proofs of Catalan's results. 
D. Andr^-^ proved that, if aj, . . . , a„ have the sum N and if k of the a's 
are not di\isible by the integer >1 which divides the greatest number of 
the a's, then (iV— A;)! is di\'isible by ai!. . .a„!. 
J. Bourguet^^ proved that, if k^2, 
{kmi)l ikmo)\...{knh)\ 
— , . ^ , ; r: = mteger. 
Wi!. . .nikl (wi+. . .+mk)l 
M. WeilP^ proved that the multinomial coefficient (tq) ! -r- {q\y is divisible 
by tl 
WeilP^ stated that the following expression is an integer : 
(a+i8+ ■ • ■ +pg+Pigi+ ■ ■ • +rst) ! 
am. . .{piyq\{p,\y^q,\. . .{r\ns\yt\' 
WeilP^ stated the special case that {a-\-^-\-pq-\-rs)\ is divisible by 
a\^\{q\rp\{s\yrl 
D. Andr^-^ proved that (tq) ! -r- (g!)' is divisible by (<!)* if for every prime 
p the sum of the digits of q to base p is ^k. 
Ch. Hermite'^^ proved that n! divides 
m{7n+k){m+2k) . . . lm+{n-l)k]k''-\ 
C. de PoUgnac^" gave a simple proof of the theorem by WeilP^ and 
expressed the generalization by Andr^^^ in another and more general form. 
E. Catalan^^ noted that, if s is the number of powers of 2 having the sum 
^+^' (2a)! (26)! 
a!6!(a+6)! 
is an even integer and the product of 2' by an odd number. 
E. Catalan^^ noted that, if n = a+6+ . . . -{-t, 
n\{n+t) 
a\h\...tl 
is divisible by a+t, h+t,. . ., a-\-h-\-t,. . ., a-\-b+c+t,. . .. 
E. Ces^ro^^ stated and Neuberg proved that (p) is divisible by n(n — 1) 
if p is prime to n(n — 1), and p — l prime to n — 1; and divisible by (p + 1) 
X(p+2) if p-\-l is prime to n+1, and p+2 is prime to (n + l)(n+2). 
"Zeitschrift Math. Phys., 20, 1875, 161-3. Die Elemente der Zahlentheorie, 1892, 37-39. 
«Bull. Soc. Math. France, 1, 1875, 84. 
"Nouv. Ann. Math., (2), 14, 1875, 89; he wrote r(n) incorrectly for n!; see p. 179. 
'MDomptes Rendus Paris, 93, 1881, 1066; Mathesis, 2, 1882, 48; 4, 1884, 20; Lucas, Th^orie 
des nombres, 1891, 365, ex. 3. Proof by induction, Amer. M. Monthly, 17, 1910, 147. 
»«Bull. Soc. Math. France, 9, 1880-1, 172. Special case, Amer. M. Monthly, 23, 1916, 352-3. 
"Mathesis, 2, 1882, 48; proof by Li6nard, 4, 1884, 20-23. 
"Comptes Rendus Paris, 94, 1882, 426. 
"Faculty des Sc. de Paris, Cours de Hermite, 1882, 138; ed. 3, 1887, 175; ed. 4, 1891, 196. 
Cf. Catalan, M6m. Soc. Sc. de Li6ge, (2), 13, 1886, 262-^ ( = Melanges Math.); Heine."" 
»"Comptes Rendus Paris, 96, 1883, 485-7. Cf . Bachmann, Niedere Zahlentheorie, 1, 1902. 59-62. 
«Atti Accad. Pont. Nouvi Lincei, 37, 1883-4, 110-3. 
"Mathesis, 3, 1883, 48; proof by Cesiro, p. 118. 
"Ibid., 5, 1885, 84. 
