CHAP. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 271 Hence, by Legendre's formula (1),
Insert the value of a+/3 obtained by adding equations (3). Thus
A. Genocchi72 proved that, if m is the sum of n integers a, 6, . . . , k, each divisible by p — 1, and if m<pn— 1, then w!-r {<z!&!. . .&!} is divisible by the prime p.
J. Wolstenholme73 proved that (2;Hi) = l(mod n3) if n is a prime > 3.
H. Anton4 (303-6) proved that if n = vp+a, r = wp+b, where a, 6, v, w are ail less than the prime p,
according as a^6 or a<b. M. Jenkins730 considered for an odd prime p the sum
extended over all the integers k between nr/(p—1) and —mr/(p —1), inclusive, and proved that ov=o-p (mod p) if the g. c. d. of r, p —1 equals that of p, p-1.
E. Catalan74 noted that (n£l})=l(mod p), if p is a prime.
Ch. Hermite75 proved by use of roots of unity that the odd prime p divides
(2n+l\.(2n+l\.(2n+l\
E. Lucas76 noted that, if m = pm!+Mj n=pui+v, ju<p, v<p, and p is a prime,
In general, if nlt fj.2, • • • denote the residues of m and the integers contained in the fractions ra/p, m/p2, . . . , while the v's are the residues of n, [n/p], . . . ,
... (modp). E. Lucas77 proved the preceding results and
according as n is between Oand p, 0 and p — 1, or 1 and p.
72Nouv. Ann. Math., 14, 1855, 241-3.
73Quar. Jour. Math., 5, 1862, 35-9. For mod. n2, Math. Quest. Educ. Times, (2), 3, 1903, 33. 73«Math. Quest. Educ. Times, 12, 1869, 29. 74Nouv. Corresp. Math., 1, 1874-5, 76.
76Jour. fur Math., 81, 1876, 94. 76Bull. Soc. Math. France, 6, 1877-8, 52.
77Amer. Jour. Math., 1, 1878, 229, 230. For the second, anon.43 of Ch. Ill (in 1830).