CHAP. Hi] GENERALIZATIONS OF WILSON'S THEOREM. 91
H. F. Scherk203 proved Jacobins theorem and the following: Form the sum Pnh of the multiplicative combinations with repetitions of the hth class of any n numbers less than the prime p, and the sum of the combinations without repetitions out of the remaining p~n— 1 numbers <p', then the sum or the difference of the two is divisible by p according as h is odd or even,
Let Cj denote the sum of the combinations with repetitions of the ftth class of 1, 2, . . ., fc; A\ the sum without repetitions. If Q<h<p— 1,
CjsO (mod p), j-p-*,. . ., P-2; C*p+fcsCJ.
For fc=p~-l, <%£k=n+l for /c = l,..., p. For /i = w(p-l)-K CJsCi when k<p+l. For !<&<&, the sum of Chk and At is divisible by jb«(A;+l)2; likewise, each Cand-A if ^ is odd. For fc<2fc, CJ-AJ is divisible by 2&+1. The sum of the 2nth powers of 1, . . ., k is divisible by 2A+1.
K. Hensel204 has given the further generalization: If alf . . . , an, b1} . . . , 6, are n+v=p — 1 integers congruent modulo p to 1, 2, . . . , p — 1 in some order, and
tf(x) = (x-W . . . (z- &f) =x*-Blx*~l+ . . .±BV,
then, for any j, Pn;=(-lX°£y0 (mod p), where j0 is the least residue of j mod p— 1 and BA = 0 (/c>t;).
For Steiner's Xn, Xn\l/(x)^xp~~l-l (mod p). Multiply (1) by a^c"-1-!). Thus
. - - (mod p).
Replace ^(x) by its initial expression and compare coefficients. Hence
Taking v=j = p — 2 and choosing 2,..., p — 1 for 61;...; &„, we get 1S5_(p_l)! (modp).
CONVERSE OF FERMAT'B THEOREM.
In a Chinese manuscript dating from the time of Confucius it is stated erroneously that 2n~1 — 1 is not divisible by n if n is not prime (Jeans220).
Leibniz in September 1680 and December 1681 (Mahnke,7 49-51) stated incorrectly that 2n — 2 is not divisible by n if n is not a prime. If n = rs, where r is the least prime factor of n, the binomial coefficient (?) was shown to be not divisible by n, since n— 1,. . ., n— r+1 are not divisible by r, whence not all the separate terms in the expansion of (l + l)n — 2 are
JMUeber die Theilbarkeit der Combinationssummen aus den natiirlichen Zahlon durck Prim-
zahlen, Progr., Bremen, 1864, 20 pp. »MArchiv Math. Phye,, (3), 1, 1901, 319; Kroaecker'a Zahlentheorie 1, 1901, 503.