266 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. DC P. Bachmann22 gave arithmetical proofs of Catalan's results. D. Andre*23 proved that, if ai, . . . , an have the sum N and if k of the a's are not divisible by the integer > 1 which divides the greatest number of the a's, then (N—k) I is divisible by c^L . .aj. J. Bourguet24 proved that, if k*£2, ,,...k . - : - r-. - ; - ; - r-j = integer. mil. . .mkl (T»I+ . . . +mk) \ e M. Weill25 proved that the multinomial coefficient (tq) I -5- (q\Y is divisible by tl Weill26 stated that the following expression is an integer: . +pq+piqi+ . . . Weill27 stated the special case that (a+jS+pg+rs)! is divisible by D. Andre*28 proved that (tq) I -Kg!)' is divisible by (ti)k if for every prime p the sum of the digits of q to base p is §: k. Ch. Hermite29 proved that n\ divides m(m+k)(m+2k) . . . {ro+(n-l)fc}fc11-1. C. de Polignac30 gave a simple proof of the theorem by Weill26 and expressed the generalization by Andre*28 in another and more general form. E. Catalan31 noted that, if s is the number of powers of 2 having the sum a+b> (2a)!(2&)! is an even integer and the product of 2* by an odd number. E. Catalan32 noted that, if n = a+6+.:. +t, nl(n+t) a! &!...*! is divisible by a+t, b+t,..., a+b+t,..., a+b+c+t,---- E. Ces&ro33 stated and Neuberg proved that (J) is divisible by n(n—1) if p is prime to w(n —1), and p —1 prime to n — 1; and divisible by (p+1) X (p+2) if p-fl 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 Zahlentbeorie, 1892, 37-39. »BuU. Soc. Math. France, 1, 1875, 84. 24Nouv. Ann. Math., (2), 14, 1875, 89; he wrote r(n) incorrectly for n!; see p. 179. 26Comptes Rendua Paris, 93, 1881, 1066; Mathesis, 2, 1882, 48; 4, 1884, 20; Lucas, ThSorie 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 Lienard, 4, 1884, 20-23. 2aComptes Rendus Paris, 94, 1882, 426. "Faculty des Sc. de Paris, Coure de Hermite, 1882, 138; ed. 3, 1887, 175; ed. 4, 1891, 196. Cf. Catalan, Mem. Soc. Sc. de Liege, (2), 13,1886, 262-4 ( = Melanges Math.); Heine.29" a°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. •2MatheBis, 3, 1883, 48; proof by Cesaro, p. 118. 33/6id., 5, 1885, 84.