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Full text of "History Of The Theory Of Numbers - I"

CHAP. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS.     267 E. Catalan34 noted that
F. Gomes Teixeira36 discussed the result due to Weill.26 De Presle36 proved that
(k+l)(k+2)...(k+hl)    .
- nw -    ger>
being the product of an evident integer by (hl)]/{ll(M)1}. E. Catalan37 noted that, if n is prime to 6,
(2n-4)t
H. W. Lloyd Tanner38 proved that
(X,!. . .A»!)'fo !)*
L. Gegenbauer stated and J. A. Gmeiner39 proved arithmetically that, if rt»Z3Iia,iO,-2. . .a,-., the product
is divisible by
n rite,!)'**1-"'*
y-l   ,-i
where w, A;, n, an, . • . . , ara are positive integers. This gives Hermite's29 result by taking r = s = 1 . The case m = fc = l,s = 2, is included in the result by Weill.26
Heine39" and A. Thue40 proved that a fraction, whose denominator is kl and whose numerator is a product of k consecutive terms of an arithmetical progression, can always be reduced until the new denominator contains only such primes as divide the difference of the progression [a part of Hermit e's29 result].
F. Rogel41 noted that, if P be the product of the primes between (p — 1)/2 and p+1, while n is any integer not divisible by the prime p,
(n-l)(n-2). ..(n-p + l)P/psO (mod P). S. Pincherle42 noted that, if n is a prime,
is divisible by n and, if x is not divisible by n, by n !.   If n = lip", P is divisible
v. Ann. Math., (3), 4, 1885, 487.   Proof by Landau, (4), I, 1901, 282. "Archiv Math. Phys., (2), 2 1885, 265-8.          "Bull. Soc. Math. France, 16, 1887-8, 159.
a7M6m. Soc. Roy. Sc. Ltege, (2), 15, 1888, 111 (M61anges Math. III).    Mathesis, 9, 1889, 170. 38Proc. London Math. Soc., 20, 1888-9, 287.       "Monatshefte Math. Phys., 1, 1890, 159-162. »9<*Jour. fur Math., 45, 1853, 287-8.    Cf. Math. Quest. Educ. Times, 56, 1892, 62-63. *°Archiv for Math, og Natur., Kristiania, 14, 1890, 247-250. "Archiv Math. Phys., (2), 10, 1891, 93. 42Rendiconto Sess. Accad. Sc. Istituto di Bologna, 1892-3, 17.