Chap. Ill] 
Symmetric Functions Modulo p. 
97 
The number (1, 2, . . ., r)'* is divisible by every prime >r which occurs in 
the series n+2, n+3, . . ., n+r. 
G. ToreUi"! proved that 
(ai, . . . , GnY = (ai, . . . , a„_iy+a„{ai, . . . , a^y~\ 
(fli,. . ., o„, ^y-Cfli,. . ., fln, c)'"=(6-c)(ai,. . ., a„, 6, cy-\ 
which becomes Fergola's for ai = i (i = 0, . . ., n). Proof is given of Syl- 
vester's^^^ theorem and the generahzation that >Sy,i is divisible by (}+!). 
Torelli^^^ proved that the sum o-„, « of all products of n equal or distinct 
numbers chosen from 1, 2, . . ., m is divisible by (n+T), and gave recursion 
formulas for o-n, m- 
C. Sardi^'^^ deduced Sylvester's theorem from the equations Ai — (|),, . . 
used by Lagrange. ^^ Solving them for Ap = Sp,n, we get 
pi{-iy^%,,= 
-1 
G) 
(a) 
C) 

-2 
i;--i) 

V 2 ; 

\ 3 ; 

("to 
( 
n-p+2\ /n+1 
2 y Vp+^ 
D 
If n+l is a prime we see by the last column that >S„_i.„ is divisible by n+1. 
When p = n — l, denote the determinant by D. Then if n+l is a prime, 
D is evidently divisible by n+l. Conversely, if D is divisible by n+l and 
the quotient by (n — 1) !, then n+l is a prime. It is shown that 
p=l 
r„ = F+...+n^ 
Using this for m = 1, . . . , n, we see that Vp is divisible by any integer prime 
to 2, 3, . . ., p+1 which occurs in n+l or n. Hence if n+l is a prime, it 
divides ri, . . ., r„_i, while rn=n (mod n+l). If n+l divides r„_i it is a 
prime. 
Sardi"^ proved Sylvester's theorem and the formula 
S ( — l)''>Si,, n+r-lO'k-r, n+r = 0, 
r-0 
stated by Fergola."^ 
"iQiornale di Mat., 5, 1867, 110-120. 
"276id., 250-3. 
"mid., 371-6. 
"*Ibid., 169-174. 
"Ubid., 4, 1866, 380. 
