268 HlSTOBY OF THE THEORY OF NUMBERS. [CHAP. IX by n! if and only if divisible by lip*"1"*, where ft is the exponent of the power of p dividing (n — 1)!. G. Bauer43 proved that the multinomial coefficient (n+nl+n2+ . . .)l -7- {nlttjj . . . } is an integer, and is even if two or more n's are equal. E. Landau44 generalized most of the preceding results. For integers aH) ha, each ^ 0, and positive integers %, set Then / is an integer if and only if t for all real values of the % for which 0^ a?/S !• A new example is S u& S V t»l i-l 2w+n) ! (m+2n) I " P. A. MacMahon45 treated the problem to find all a's for which /n+2\g« 2 / " /n+l\gi/ V 1 M is an integer for all values of n\ in particular, to find those "ground forms'1 from which %11 the forms may be generated by multiplication. For w = 2, the ground forms have (ax, a2) = (1, 0) or (1, 1). For m = 3, the additional ground forms are (1, 1, 1), (1, 2, 1), (1, 3, 1). For m = 4, there are 3 new ground forms; for m = 5, 13 new. J. W. L. Glaisher46 noted that, if Bp(x) is Bernoulli's function, i. e., the polynomial expression in x for lp~1-|-2p~1+ . . . + (x— l)p~l [Bernoulli150* 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). Glaisher47 noted that, if r is not divisible by the odd prime p, and Z(r+0(2r+I)...{(p-l)r+I}/ps-frA] +*} (mod p), where [^/p]r denotes the least positive root of px==t (mod r). The residues mod p3 of the same product £(r-H) . . . are found to be complicated. E. Maillet48 gave a group of order tl(q\Y contained in the symmetric group on tq letters, whence follows Weill's25 result. ^Sitzungsber. Ak. Wiss. Miinchen (Math.), 24, 1894, 346-8. "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. <8Proc. London Math. Soc., 32, 1900, 172. 47A/T0a00r,^^T. AAo+V. Qrt 1 Q/VL.1 71-Q9