90 HlSTOEY OP THE THEORY OF NUMBERS. [CHAP. Ill
FURTHER GENERALIZATIONS or WILSON'S THEOREM; RELATED PROBLEMS.
J. Steiner200 proved that, if Ak is the sum of all products of powers of ab a2,. . ., ap_fc of degree kt and the a's have incongruent residues 5^0 modulo p, a prime, then Aif . . ., Ap-.2 are divisible by p.
He first showed by induction that
A2=al2+a1a2+ . . . For example, to obtain a;3 he multiplied the respective terms of
by xt (#—03)4-03, (x—a2)+a2, (x—a^+ai. Let %,..,, ap_i have the residues 1,. . ., p—1 in some order, modulo p. For 2— a2 divisible by p, x?~1=^p_1=af~1 (mod p), so that Ap,2Xi and hence also Ap_2 is divisible by p. Then for z=a3, J.P_3X2 and Ap_z are divisible by p. For #=0, &! = !, the initial equation yields Wilson's theorem.
C. G. J. Jacobi201 proved the generalization: If aiy . . ., an have distinct residues 5^0, modulo p, a prime, and Pnm is the sum of their multiplicative combinations with repetitions m at a time, Pnm is divisible by p for m = p — n, p-n4-l,...,p-2.
Note that Steiner^s ^ is Pp-ktk- We have
i i p P »»
-1 -^nl rni _ -irv
.(%-«»), 0= Sa?/Z
j-i
Let 7i+m~l=A;+/3(p-l). Then aj+*-1=a/t (mod p). Hence if A;<n-l,
DL . .DJP^D,. . .DnSa?/A-, P«^0 (mod p).
The theorem follows by taking /3 = 1 and fc~ 0, 1, . . . , n — 2 in turn.
H. F. Scherk202 gave two generalizations of Wilson's theorem. Let p be a prime. By use of Wilson's theorem it is easily proved that
where x is an integer such that px=l (mod n!). Next, let Ckr denote the sum of the products of 1, 2, . . . , k taken r at a time with repetitions. By use of partial fractions it is proved that
(p-r-1) !(£_,_! -H-l)r==0 (mod?) (r<p-l).
It is stated that
wjour. fur Math., 13, 1834, 356; Werke 2, p. 9. "I/bid., 14, 1835, 64-5; Werke 6, 252-3.
202Bericht uber die 24. Versammlung Deutscher Naturforscher und Aerzte in 1846, Kiel, 1847, 204-208.