CHAP, ill] SYMMETRIC FUNCTIONS MODULO p. 101
K. Hensel297 proved by the method of Poinsot255 that any integral symmetric function of degree v of 1, ..., p— 1 with integral coefficients is divisible by the prime p if v is not a multiple of p — 1.
W. F. Meyer298 gave the generalization that, if ab . . . , ap_! are incongru-ent modulo pn, and each a/"1 — 1 is divisible by pn, any integral symmetric function of degree v of al} . . . , ap_i is divisible by pn if v is not a multiple of p~l. Of the (j>(pn) residues modulo pn, prime to p, there are p*(p — I)2 for which ap~l — l is divisible by p71",1"*, but by-no higher power of p, where & = !,..., n— 1; the remaining p — 1 residues give the above ait . . ., ap_!.
J. W. Nicholson299 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 ss — 1 (mod p) if n=p— 1.
G. Wertheim300 proved the same result by use of a primitive root.
A. Aubry301 took z = l, 2,. . ., p-1 in
(x+l)n-xn=nxn~l+Axn~2+ . . . +Lx+l and added the results. Thus
pn =nsn
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. Concina302 proved that sn 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 ju$ = p — 1 . The At distinct residues r{ of nth powers modulo p are the roots of xp=l (mod p), whence 2^=0 (mod p) for n not divisible by p— 1. For each rit xnzz=ri has 6 incongruent roots. Hence sn=6Sr^O. He proved also that, if p+1 is a prime >3, and n is even and not divisible by p, ln+2n+ . . . +(p/2)n is divisible by p+1.
W. H. L. Janssen van Raay803 considered, for a prune p> 3,
A -(P-1)' R _(P~1)1
Ah~~h~' a"~h(^K)
and proved that B1+J32+ • • • +B(t-u/2 is divisible by p, and
are divisible by p2.
U. Concina304 proved that S = l+2n+. . . +kn is divisible by the odd number k 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, S is divisible by k or only by k/2 according
297Archiv Math. Phys., (3), 1, 1901, 319. Inserted by Hensel in Kronecker'a Vorlesungen tiber
Zahlentheorie I, 1901, 104-5, 504.
298Archiv Math. Phys., (3), 2, 1902, 141. Cf. Meissner" of Ch. IV. 298Amer. Math. Monthly, 9, 1902, 212-3. Stated, 1, 1894, 188. 800Anfangsgriinde der Zahlentheorie, 1902, 265-6. '"L'enseignement math., 9, 1907, 296. »MPeriodico di Mat., 27, 1912, 79-83. «»Nieuw Archief voor Wiskunde, (2), 10, 1912, 172-7. 804Periodico di Mat., 28, 1913, 164-177, 267-270.