History Of The Theory Of Numbers - I

CHAP. IX] DIVISIBILITY OF FACTOEIALS, MULTINOMIAL COEFFICIENTS. 273 C. Szily88 noted that no prime >2a divides
iCO'
and specified the intervals in which its prime factors occur.
F. Morley89 proved that, if p = 2n+l is a prime, (2nn)-(-l)n24n is divisible by p3 if p>3. That it is divisible by p2 was stated as an exercise in Mathews' Theory of Numbers, 1892, p. 318, Ex. 16.
L. E. Dickson90 extended Kummer's71 results to a multinomial coefficient
M and noted the useful corollary that it is not divisible by a given prime p
if and only if the partition of m into mi,..., mt arises by the separate
partition of each digit of m written to the base p into the corresponding
. digits of mi,..., mt. In this case he proved that
n CL'^
Mz=Tl (1) *' «j| (mod p), mfc=a0(*)pn+.
This also follows from (2) and from
(mod p).
F. Mertens91 considered a prime p^n, the highest powers p* and 2" of p and 2 which are ^n, and set na=[n/2a]. Then ra!-5- {ni\n2l. . .nv\} is divisible by lip*", where p ranges over all the primes p.
J. W. L. Glaisher92 gave Dickson's90 result for the case of binomial coefficients. He considered (349-60) their residues modulo pn, and proved (pp. 361-6) that if (ri)r denotes the number of combinations of n things r at a time, £(n)r=E(j)* (mod p), where p is any prime, n any integer ==.? (mod p — 1), while the summation extends over all positive integers r, r^n, r=k (mod p — 1), and j, k are any of the integers 1,. . ., p — 1. He evaluated 2J[(n)r-f-p] when r is any number divisible by p — 1, and (n)r is divisible by p, distinguishing three cases to obtain simple results.
Dickson93 generalized Glaisher's92 theorem to multinomial coefficients: Let k be that one of the numbers 1, 2, . . ., p — 1 to which m is congruent modulo p — 1, and let kly . . ., kt be fixed numbers of that set such that ki+ . . . 4-k^k (mod p — 1). Then if p is a prime,
« / \ I (A/i. A/9) . . .j Kt) II A/i"T" • • • \ K/i^^ fa
The second of the two proofs given is much the simpler.
s«Nouv. Ann. Math., (3), 12, 1893, Exercices, p. 52.* Proof, (4), 16, 1916, 39-42.
89Annals of Math., 9, 1895, 168-170.
MIbid., (1), 11, 1896-7. 75-6: Quart. Jour. Math., 33, 1902, 378-384.
MSitzungsber. Ak. Wiss. Wien (Math.), 106, lla, 1897, 255-6.
92Quar. Jour. Math., 30, 1899, 150-6, 349-366.
»*Ibid., 33, 1902, 381-4.