Chap. Ill] SyMMETKIC FUNCTIONS MODULO p. 101 
K. HenseP^^ proved by the method of Poinsot^^^ that any integral sym- 
metric function of degree v of 1,..., p — 1 with integral coefficients is 
divisible by the prime p if y is not a multiple of p — 1. 
W. F. Meyer^^^ gave the generalization that, if ai, . . . , ap_i are incongru- 
ent modulo p", and each af~^ — 1 is divisible by p", any integral symmetric 
function of degree voiai,..., ap_i is divisible by p'* if v is not a multiple of 
p — 1. Of the </)(p") residues modulo p", prime to p, there are p'^Cp — 1)^ for 
which a^~^ — l is divisible by p"~^~'^, but by no higher power of p, where 
A; = 1, . . ., n— 1; the remaining p — 1 residues give the above ai,..., a^-i. 
J. W. Nicholson^^^ noted that, if p is a prime, the sum of the nth powers 
of p numbers in arithmetical progression is divisible by p if n<p — 1, and 
= — 1 (mod p) if 71 = p — 1. 
G. Wertheim^"'' proved the same result by use of a primitive root. 
A. Aubry^°^ took x = 1, 2, . . . , p — 1 in 
(a:+l)"-rc" = nx^-i+Ax"-2+ . . . -{-Lx+l 
and added the results. Thus 
p'' = ns„_i+As„_2+. . .+Lsi+p. 
Hence by induction Sn_i is divisible by the prime p if n<p. He attributed 
this theorem to Gauss and Libri without references. 
U. Concina^°^ proved that s„ is divisible by the prime p>2 if n is not 
divisible by p — 1. Let 5 be the g. c. d. of n, p — 1, and set ^i5 = p — 1. The 
)u distinct residues Ti of nth powers modulo p are the roots of ^"=1 (mod p), 
whence Sr,=0 (mod p) for n not divisible by p — 1. For each r^-, x'*=rj has 
6 incongruent roots. Hence s„=5Sri=0. He proved also that, if p+1 is 
a prime >3, and n is even and not divisible by p, l''+2''+ . . . +(p/2)'* is 
divisible by p+1. 
W. H. L. Janssen van Raay^°^ considered, for a prime p>3, 
(p-1)! ^ (P-I)I 
^''~ h ' ^'~h{v-h) 
and proved that B^-\-B2-\- . . . +-B(p_i)/2 is divisible by p, and 
are divisible by p^. 
U. Concina^o^ proved that ^ = 1+2"+ . . .+/c" is divisible by the odd 
number A; if n is not divisible by p — 1 for any prime divisor of p of k. Next, 
let k be even. For n odd > 1, ^ is divisible by k or only by k/2 according 
"'Archiv Math. Phys., (3), 1, 1901, 319. Inserted by Hensel in Kronecker's Vorlesungen tiber 
Zahlentheorie I, 1901, 104-5, 504. 
"sArchiv Math. Phys., (3), 2, 1902, 141. Cf. Meissner'" of Ch. IV. 
"^Amer. Math. Monthly, 9, 1902, 212-3. Stated, 1, 1894, 188. 
^ooAnfangsgninde der Zahlentheorie, 1902, 265-6. 
'"iL'enseignement math., 9, 1907, 296. 
»°2Periodico di Mat., 27, 1912, 79-83. 
"'Nieuw Archief voor Wiskunde, (2), 10, 1912, 172-7. 
so^Periodico di Mat., 28, 1913, 164-177, 267-270. 
