CHAP. Ill]
SYMMETRIC FUNCTIONS MODULO p.
97
The number (1, 2,. . ., r)n is divisible by every prime >r which occurs in the series n+2, n +3,. .., n+r. G. Torelli271 proved that
(01,. . ., an)r=(a!,. . ., a«_i)r+an(ai, . . ., an)r~l, (01,. . ., an, 6)r-(ai,. . ., an, c)r=(6-c)(ai,. . ., an, 6, c)1"'1,
which becomes Fergola's for a^i (i=0,. . ., n). Proof is given of Sylvester's269 theorem and the generalization that Siti is divisible by (££}).
Torelli272 proved that the sum o-n,m of all products of n equal or distinct numbers chosen from 1, 2, . . ., m is divisible by (J+O, and gave recursion formulas for <rnt m.
C. Sardi273 deduced Sylvester's theorem from the equations Al = (?),... used by Lagrange.18 Solving them for Ap=Sp,n, we get
0
-2
0
0
-3
_i
GO GO
CO
If n+1 is a prime we see by the last column that $n_i,n is divisible by n+1. When p = n—1, denote the determinant by D. Then if n+1 is a prime, D is evidently divisible by n+1. Conversely, if D is divisible by n+1 and the quotient by (n—1)!, then n+1 is a prime. It is shown that
0 /n+l\ V 2 )
o /n+l\
v 3 ;
0 e:1)
2 J 1 (n+l)
Using this for w = l,..., n, we see that rp is divisible by any integer prime to 2, 3,.. ., p+1 which occurs in n+1 or n. Hence if n+1 is a prime, it divides r1;..., rn_1; while rn=n (mod n+1). If n+1 divides rn_x it is a prime.
Sardi274 proved Sylvester's theorem and the formula
stated by Fergola.275
J71Giornale di Mat., 5, 1867, 110-120. "'/bid., 250-3. »7s/6td., 371-6. ™Ibid., 169-174.