Chap. Ill] Genekalizations OP Wilson's Theorem. 91 
C;.r-iCr-' + {-iy^O, C:,-m\^0 (modp), ^ = ^- 
H. F. Scherk^''^ proved Jacobi's 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 Cl denote the sum of the combinations with repetitions of the hth 
class oi 1, 2,. . ., k; Al the sum without repetitions. If 0<h<p — l, 
Ci^O (mod p), j = p-k,.. ., p-2; Cl+,^Cl 
For h = p-l, Ci;ik=n+1 for k = l,..., p. For h = m{p-l)+t, Cl=Ci 
when k<p+l. For l<h<k, the sum of Cl and A^ is divisible by 
A;^(fc+1)^; likewise, each C and A if /i is odd. For h<2k, Cl—Al is divisible 
by 2A:H-1. The sum of the 2nth powers of 1, . . . , /c is divisible by 2k-\-l. 
K. HenseP'^ has given the further generalization: If ai, . . . , a,^, 61, . . . , 6, 
are n-\-v = p — l integers congruent modulo p to 1, 2, . . . , p — 1 in some order, 
and 
^l/{x) = (x-b,)... (rr-6J =x'-B,x'-'+ . . .=^B„ 
then, for any j, Pnj^i — iy'Bj^ (mod p), where jo is the least residue of j 
mod p — 1 and Bh = {k>v). 
For Steiner's Z„, Z„^(a;)=a:^-i-l (mod p). Multiply (1) by 
a:"(x^-^-l). Thus 
X''rP{x)^X'-'+PnlX^-'+ . . .+Pnp-2X + Pnp-l-l + ^"^~^"^ 
X 
+ ^"^^^"^"' +... (modp). 
X 
Replace \f/{x) by its initial expression and compare coefficients. Hence 
p^j^i-iyBj{j=i,...,v). 
Taking v=j = p — 2 and choosing 2,..., p — 1 for 61,..., 6„ we get 
1= — (p — 1)! (mod p). 
Converse of Fermat's Theorem. 
In a Chinese manuscript dating from the time of Confucius it is stated 
erroneously that 2""^ — 1 is not divisible by n if n is not prime (Jeans^^^). 
Leibniz in September 1680 and December 1681 (Mahnke,^ 49-51) stated 
incorrectly that 2'*— 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 (1 + 1)" — 2 are 
^o^Ueber die Theilbarkeit der Combinationssummen aus den natiirlichen Zahlen durch Prim- 
zahlen, Progr., Bremen, 1864, 20 pp. 
^"Archiv Math. Phys., (3), 1, 1901, 319; Kronecker's Zahlentheorie 1, 1901, 503. 
