History Of The Theory Of Numbers - I

270 HISTOKY or THE THEORY OF NUMBERS. [CHAP, ix
a, a+N,. . ., a+Nn, we conclude that the product of their differences is divisible by n \ (n — 1) ! . . . 2 ! = v. But the product equals
p=(j\T-l)n-1 (2V2-!)"-2. . .(N^-inN*-1-!),
multiplied by a power of N. Hence, if N is prime to nl, P is divisible by v\ in any case a least number X is found such that A7"XP is divisible by v. It is shown that the product of the differences of ml7. . ., mk is divisible by kl(k — 1)1. . .21 if there be any integer p such that m^p, . . ., mk+p are relatively prime to each of 1, 2, . . . , k. It is proved that the product of any n distinct integers multiplied by the product of all their differences is a multiple of n!(n-l)I. . .2!.
E. de Jonquieres63 and F. J. Studnicka64 proved the last theorem.
E. B. Elliott65 proved Segar's theorem in the form: The product of the differences of n distinct numbers is divisible by the product of the differences of 0, 1,..., n— 1. He added the new theorems: The product of the differences of n distinct squares is divisible by the product of the differences of O2, I2,..., (n—l}2] that for the squares of n distinct odd numbers, multiplied by the product of the n numbers, is divisible by the product of the differences of the squares of the first n odd numbers, multiplied by their product.
RESIDUES OF MULTINOMIAL COEFFICIENTS.
Leibniz4'7 of Ch. Ill noted that the coefficients in (Sa)p~Sap are divisible by p.
Ch. Babbage69 proved that, if n is a prime, (2n-Ti) — 1 is divisible by n2, while (p£n) — 1 is divisible by p if and only if p is a prime.
G. Libri70 noted that, if m — 6p-\-l is a prime,
»-(»)!+(*)*-....*
E. Kummer71 determined the highest power pN of a prime p dividing ' . . . +atpl, B=bQ+b1p+ . . . +btpl,
where the at- and 6t belong to the set 0', 1, . . ., p — 1. We may determine Ci in this set and e»- = 0 or 1 such that
(3)
Multiply the first equation by 1, the second by p} the third by p2, etc., and add. Thus
^Comptes Rendus Paris, 120, 1895, 408-10, 534-7.
"Vestnik Ceske Ak., 7, 1898, No. 3, 165 (Bohemian).
"Messiager Math., 27, 1897-8, 12-15.
"Edinburgh Phil. Jour., 1, 1819, 46.
70Jour. fur Math., 9, 1832, 73. Proofs by Stern, 12, 1834, 288.
nlbid., 44, 1852, 115-6. Cayley, Math. Quest. Educ. Times, 10, 1868, 88-9.