90 History of the Theory of Numbers. [Chap, hi 
Further Generalizations of Wilson's Theorem; Related Problems. 
J. Steiner-°° proved that, if Ak is the sum of all products of powers of 
Gi, 02,..., Op-it of degree k, and the o's have incongruent residues p^O 
modulo p, a prime, then A^,. . ., Ap_2 are divisible by p. 
He first showed by induction that 
z =2i.p^i-\-AiA.p^2~T' ■ ■ ■ \Ap-2-^i\Ap^i, 
Xk={x-ai). . .{x-ttk), ^i = Oi+. . .+ap_i, 
^2 = 01^ + 0102+ . . . +0iOp_2 + O2^+O2O3+ . . . +oJ_2, 
For example, to obtain x^ he multipUed the respective tenns of 
V=(X — Oi)(x — 02) + (01+02) (x — 0i)+0i^ 
by X, (a:-03)+a3, (x— 02)+02, (x — oO+Oi. Let Oi,..., Op_i have the 
residues 1,. . ., p — l in some order, modulo p. For x — 02 divisible by p, 
x^~^=Ap_i = a{~^ (mod p), so that Ap_2Xi and hence also Ap_2 is divisible 
by p. Then for x=a3, ^p_3X2 and ^p_3 are divisible by p. For x = 0, 
ai = l, the initial equation yields Wilson's theorem. 
C. G. J. Jacobi"''^ proved the generahzation : If Oi,. . ., a„ have distinct 
residues f^ 0, modulo p, a prime, and Pr^m is the sum of their multipUcative 
combinations with repetitions m at a time, Pnm is divisible by p for w = p— n, 
p-n+1,..., p-2. 
Note that Steiner's Ah is Pp-k.k- We have 
][ J^ I Pfil ■ Pn2 
(x-oi) . . . (x-o„) "x""^x"+i^x"+2"^ • • •' ' """,=; 
(1) 7:r^7v^^^^.=i+9h+^2+--: P..= :^a;^-'/Dj, 
Dj = {cLj - Oi) . . . (Oj - a,_i) (oy - Oy+i) . . . (o, - J , 0=2 Oy /D, {k<n-l). 
j=i 
Let n+m-l = k+^{p-\). Then o^+^-^^o/ (mod p). Hence if 
A;<Cn — 1 
' Di . . .Dr,Pn,„=D, . . .Dj:a)/D„ P^^^O (mod p). 
The theorem follows by taking /3 = 1 and ^' = 0, 1, . . ., n— 2 in turn. 
H. F. Scherk^°- gave two generaUzations of Wilson's theorem. Let p be 
a prime. By use of Wilson's theorem it is easily proved that 
n! 
where x is an integer such that px=l (mod n!). Next, let C/ denote the 
sum of the products of 1, 2, . . . , ^ taken r at a time with repetitions. By use 
of partial fractions it is proved that 
(p-r-l)!C;_,_i+(-l)'-=0(modp) (r<p-l). 
It is stated that 
""Jour, fiir Math., 13, 1834, 356; Werke 2, p. 9. 
""/bid., 14, 1835, 64-5; Werke 6, 252-3. 
'"Bericht iiber die 24. Versammlung Deutscher Naturforscher und Aerzte in 1846, Kiel, 
1847, 204-208. 
(p-n-l)!^(-ir^^(modp), 
