268 History of the Theory of Numbers. [Chap, ix 
by n ! if and only if divisible by IIp"'''^, where /3 is the exponent of the power 
of p dividing (n — 1)!. 
G. Bauer^^ proved that the multinomial coefficient (n+ni+n2+. . .)' 
-7- {7i!7?i! . . . } is an integer, and is even if two or more n's are equal. 
E. Landau^^ generalized most of the preceding results. For integers 
Qij, bij, each ^ 0, and positive integers Xj, set 
Then / is an integer if and only if 
m n 
t=i t=i 
for all real values of the Xj for which O^Xj^l. A new example is 
(4m)!(4n)! a 
m!n!(2m+n)!(m+2n)!~^^^^^^' | 
P. A. MacMahon^^ treated the problem to find all a's for which 
is an integer for all values of n; in particular, to find those "ground forms" 
from which all the forms may be generated by multiplication. For m = 2, 
the ground forms have (ai, a2) = (1, 0) or (1, 1). For m = 3, the additional 
ground forms are (1, 1, 1), (1, 2, 1), (1, 3, 1). For ?7i = 4, there are 3 new 
ground forms; for m = 5, 13 new. 
J. W. L. Glaisher^® noted that, if Bp{x) is Bernoulli's function, i. e., the 
polynomial expression in x for F~^+2^"^+ . . . + (x — 1)^"^ [Bernoulli^^"'' 
of Ch. V], 
x{x-\-l) . . .{x-\-p — l)/p=Bp{x)—x (mod p). 
He gave (ibid., 33, 1901, 29) related congruences involving the left member 
and Bp_i{x). 
Glaisher^^ noted that, if r is not divisible by the odd prime p, and 
l = kp+t, 0^t<p, 
l{r+l){2r+l) . . . {(p-l)r+i)/p^-|[^]^+A:} (mod p), 
where [t/p]r denotes the least positive root of px=t (mod r). The residues 
mod p^ of the same product l{r-\-l) . . . are found to be complicated. 
E. Maillet*^ gave a group of order t\{q\y contained in the sjrmmetric 
group on tq letters, whence follows Weill's^^ result. 
«SitzunKsber. Ak. Wiss. Miinchen (Math.), 24, 1894, 34&-8. 
"Nouv. Ann. Math., (3), 19, 1900, 344-362, 576; (4), 1, 1901, 282; Archiv Math. Phys., (3), 1, 
1901, 138. Correction, Landau." 
«Trans Cambr. Phil. Soc, 18, 1900, 12-34. 
"Proc. London Math. Soc, 32, 1900, 172. 
^'Messenger Math., 30, 1900-1, 71-92. 
*»Mem. Pr6s. Ac. Sc. Paris, (2), 32, 1902, No. 8, p. 19. 
