Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 201 
This formula is simplified in the case tx = \l/{m) and the numbers belonging 
to this Haupt-exponent are called primitive roots of m. The primitive 
roots of m divide into families of 0(i/'(m)) each, such that any two of one 
family are powers of each other modulo m, while no two of different families 
are powers of each other. Each family is subdivided. In general, not 
every integer prime to m occurs among the residues modulo m of the powers 
of the various primitive roots of m. 
A. Cunningham"^ considered the exponent ^ to which an odd number q 
belongs modulo 2"*; and gave the values of ^ when m^ 3, and when q = 2^12='= 1 
(fi odd), m>3. When g' = 2''=Fl and m>x-\-l, the residue of q^^^^ can 
usually be expressed in one of the forms 1=f2", 1=f2"=f2^. 
G. Fontene"^ determined the numbers N which belong to a given 
exponent p"'~''8 modulo p"", where 5 is a given divisor of p — l, and h^l, 
without employing a primitive root of p"". li p>2, the conditions are that 
N shall belong to the exponent 8 modulo p and that the highest power of 
p dividing N^ — 1 shall he p^, l^h^m. 
*M. Demeczky"^ discussed primitive roots. 
E. Landau"^ proved the existence of primitive roots of powers of odd 
primes, discussed systems of indices for any modulus n, and treated the 
characters of n. 
G. A. Miller"^ noted that the determination of primitive roots of g 
corresponds to the problem of finding operators of highest order in the 
cyclic group G of order g. By use of the group of isomorphisms of G it is 
shown that the primitive roots of g which belong to an exponent 2q, where 
q is an odd prime, are given by —a", when a ranges over those integers 
between 1 and g/2 which are prime to g. As a corollary, the primitive 
roots of a prime 2g+l, where q is an odd prime, are — a^, l<a<g+l. 
A. N. Korkine"^ gave a table showing for each prime p<4000 a primitive 
root g and certain characters which serve to solve any solvable congruence 
x^=a (mod p), where g is a prime dividing p — l. Let q" be the highest 
power of q dividing p — 1. The characters of degree q are the solutions of 
M« = l, u'' = u, u"' = u',..., (w^"-")' = w(''-2) (mod p) 
and hence are the residues of the powers of g^p~'^^^^ for k = l,. . ., a. There 
are noted some errors in the Canon of Jacobi^^ and the table of Burckhardt. 
Korkine stated that if p is a prime and a belongs to the exponent e = {p — 1)/5, 
exactly (f){p — l)/<l>{e) of the roots of a;* = a (mod p) are primitive roots of p. 
K. A. Posse"^ remarked that Korkine constructed his table without 
knowing of the table by Wertheim,^^and extended Korkine's tables to 10000. 
"^Messenger of Math., 37, 1907-8, 162-4. 
"*Nouv. Ann. Math., (4), 8, 1908, 193-216. 
"^Math 6s Phys. Lapok, Budapest, 17, 1908, 79-86. 
ii«Handbuch . . .Verteilung der Primzahlen, I, 1909, 391-414, 478-486. 
ii'Amer. Jour. Math., 31, 1909, 42-4. 
iisMatem. Shorn. Moskva (Math. Soc. Moscow), 27, 1909, 28-115, 120-137 (in Russian). Cf. 
D. A. Grave, 29, 1913, 7-11. The table was reprinted by Posse."* 
iio/bid., 116-120, 175-9, 238-257. Reprinted by Posse.»" 
