Chap. Ill] SYMMETRIC FUNCTIONS MoDULO p. 95 
three primes, as well as P = 13-37-73457, for each of which the congruence 
holds for every a prime to P. 
Welsch^se stated that ifk = 4n-\-l is composite and < 1000, 2^-^ = 1 (mod 
k) only for A; = 561 and 645; hence 71**= 1 (mod k) for these two k's. 
P. Bachmann^^^ proved that x^'^~^'= 1 (mod pq) is never satisfied by all 
integers prime to pq if p and q are distinct odd primes [Carmichael^^^]. 
Symmetric Functions of 1, 2,. . .p— 1 Modulo p. 
Report has been made above of the work on this topic by Lagrange, ^^ 
Lionnet," Tchebychef,^^ Sylvester,'^^ ottmger,^^ Lucas,!"^ Cahen,!^^ Aubry,^" 
Arevalo,^^^ Schuh,^^^ Frattini,!^^ steiner,^^^ Jacobi,^^! Hensel.^o^ 
We shall denote l"+2"+ . . . +(p — I)'* by s^, and take p to be a prime. 
E. Waring^^° wrote a, j3, . . .for 1, 2, . . . , x, and considered 
s = a^^^y' . . . ^a^^^y' . . . +a"/3^7'' .... 
If < = a+6+c+ . . .is odd and <a:, andx+1 is prime, s is divisible by (x+l)^. 
If t<2x and a, 6, . . .are all even and prime to 2a; +1, s is divisible by 2a;-f 1. 
V. Bouniakowsky^^^ noted that s^ is divisible by p^, if p> 2 and m is odd 
and not =1 (mod p — 1); also if both w=l (mod p — 1) and m=0 (mod p). 
C. von Staudt252 proved that, if S,Xx) = 1+2"+ • • • +a;% 
S^{ah)=bSM+naSn-iia)Si{b-l) (mod a"), 
2S2n+i{a)^{2n+l)aS2M (mod a^). 
If a,h,. . ., I are relatively prime in pairs, 
S,Xah...l) SM S^il) 
ah. . .1 a ' ' ' I 
= integer. 
A. Cauchy253 proved that 1 + 1/2+ . . . +l/(p-l)=0 (mod p). 
G. Eisenstein^^^ noted that s^= — 1 or (mod p) according as m is or 
is not divisible by p — 1. If w, n are positive integers <p — 1, 
'iyia+iy^O or - (p_f_ J' (mod p), 
according as m+n<or^p — 1. 
L. Poinsot^^^ noted that, when a takes the values 1, . . . , p — 1, then (ax)" 
has the same residues modulo p as a", order apart. By addition, SnX'*=Sn 
(mod p). Take x to be one of the numbers not a root of a;"=l. Hence 
s„=0 (mod p) if n is not divisible by p — 1. 
^''L'mtermMiaire des math., 20, 1913, 94. 
*»'Archiv Math. Phys., (3), 21, 1913, 185-7. 
""Meditationes algebraicae, ed. 3, 1782, 382. 
">BuIl. Ac. Sc. St. P^tersbourg, 4, 1838, 65-9. 
"'Jour, fiir Math., 21, 1840, 372-4. 
"'M^m. Ac. Sc. de I'lnstitut de France, 17, 1840, 340-1, footnote; Oeuvres, (1), 3, 81-2. 
«"Jour. fiir Math., 27, 1844, 292-3; 28, 1844, 232. 
»"Jour. de Math., 10, 1845, 33-4. 
