CHAPTER IX.
DIVISIBILITY OF FACTORIALS AND MULTINOMIAL COEFFICIENTS. HIGHEST POWER OF A PRIME DIVIDING ml
Genty1 noted that the highest power of 2 dividing (2n) ! is 22n~1, and the quotient is 3n-1(5-7)n-2(9-ll-13-15)n-3(17. . .31)n~4. . .(2n-l). In general if P = 2m+2nj+. . .+2% where the n's decrease, the highest power of 2 dividing PI is 2p~r.
A. M. Legendre2 proved that if pM is the highest power of the prime p which divides m\, and if [x] denotes the greatest integer ^x,
where s = a0+ . . . +an is the sum of the digits of m to the base p :
Th. Bertram3 stated Legendre's result m an equivalent form. H. Anton4 proved that, if n=vp+a,j a<p, v<p, and p is a prime,
— ;s5(p-l)'a!t>! (modp), while, if v
-~p= (p - l)t4Va ! a ! w ! w ! (mod p) .
D. Andre*5 stated that the highest power pM of the prime p dividing n! is given explicitly by /i^Sjlfln/p*] and claimed that merely the method of finding M had been given earlier. He applied this result to prove that the product of n consecutive integers is divisible by n!.
J. Neuberg6 determined the least integer m such that ml is divisible by a given power of a prime, but overlooked exceptional cases.
L. Stickelberger7 and K. Hensel8 gave the formula [cf. Anton4].
rff} \
(2) ^E(-l)*a0!aJ...an!(modp).
F. de Brun9 wrote g[u] for the exponent of the highest power of the prime p dividing u. He gave expressions for
in terms of the functions ft (a; k) = lfc+2fc+ . . . +a*. A special case gives (1).
TOist. et M<5m. Ac. R. Sc. Inscript. et Belles Lettres de Toulouse, 3, 1788, 97-101 (read Dec. 4,
1783).
2Th<§orie des nombres, ed. 2, 1808, p. 8; ed. 3, 1830, I, p. 10. 3Einige Satze aus der Zahlenlehre, Progr. Coin, Berlin, 1849, 18 pp. 4Archiv Math. Phyp., 49, 1869, 298-9. ^Nouv. Ann. Math., (2), 13, 1874, 185.
6Mathesis, 7, 1887, 68-69. Cf. A. J. Kempner, Amer. Math. Monthly, 25, 1918, 204-10. 7Math. Annalen, 37, 1890, 321. "Archiv Math. Phys., (3), 2, 1902, 294. "Arkiv for Matematik, Astr., Fysik, 5, 1904, No. 25 (French).
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