198 History of the Theory of Numbers. [Chap, vn 
F. Mertens^- called I'l,. . ., ip the system of indices of n modulo k if 
n=gi\ . .qj" (mod A:) for the g's of Kronecker.^° Such systems of indices 
differ from Dirichlet's. 
C. Moreau^^ set A^ = pV . . . , v = p''~^q^~^ . . . , where p, q,... are distinct 
primes. Take € = 1 if iV" is not divisible by 4 or if N = 4, but e = 2 if iV is 
divisible by 4 and A'' > 4. Let \p{N) denote the 1. c. m. of v/e, p — l,q — l,. . . 
[equivalent to Cauehy's-^ maximum indicator for modulus N]. For A 
prime to N, A*^^^= 1 (mod N) . If A^ = p'', 2p^ or 4 (so that N has primitive 
roots), yp{N) =4>{N) [Lucas^^j^ ^j^^^.^ -^ ^ ^^^^^ ^^ values of A^< 1000 and 
certain higher values for which \p{N) has a given value < 100. 
A. Cunningham^"* noted that we may often abbre\iate Gauss' method 
of finding a primitive root of a prime p by testing whether or not the trial 
root a is a primitive root before computing the residues of all powers of a. 
The tests are the simple rules to decide whether or not a is a quadratic or 
cubic residue of p. If a is both a quadratic non-residue and a cubic non- 
residue of p = 3co+l, and if a^^l for every/ dividing p — 1 except /=p — l, 
then a is a primitive root. 
A. Cunningham^^ gave tables showing the residues of the successive 
powers of 2 when divided by each prime or power of prime < 1000, also 
companion tables showing the indices x of 2"" whose residues modulo p'' are 
1, 2, 3, . . .. The tables are more convenient than Jacobi's Canon-^ (errata 
given here) for the problem to find the residue of a given number with 
respect to a given power of a prime, but less convenient for finding all roots 
of a given order of a given prime. There are given (p. 172) for each power 
p^< 1000 of a prime p the factors of 0(p^"), the exponent ^ to which 2 belongs 
modulo p'', and the quotient 0/^. 
E. Cahen^^ proved that if p is a prime >(32"'^'-l)/2'"+^ and if 5 = 
2^+'^p-\-l (7«>0) is a prime, then 3 is a primitive root of q, whereas 
Tchebychef^^ had the less advantageous condition p>3^^V2'"+^. Other 
related theorems by Tchebychef are proved. There are companion tables 
of indices for primes < 200. 
G. A. Miller^^ appUed the theory of groups to prove the existence of 
primitive roots of p", to show that the primitive roots of p^ are primitive 
roots of p", and to determine primitive roots of the prime p. 
L. Kronecker^^ discussed the existence of primitive roots, defined sys- 
tems of indices and appHed them to the decomposition of fractions into 
partial fractions. He developed (pp. 375-388) the theor>^ of exponents to 
which numbers belong modulo p, a prime, by use of the primitive factor 
"Sitzungsber. Ak. Wien (Math.), 106, II a, 1897, 259. 
«Nouv. Ann. Math., (3), 17, 1898, 303. 
"Math. Quest. Educat. Times, 73, 1900, 45, 47. 
•'A Binary Canon, showing residues of powers of 2 for divisors under 1000, and indices to 
residues, London, 1900, 172 pp. Manuscript was described by author. Report British 
Assoc, i895, 613. Errata, Cunningham.'" 
"filaments de la th^orie des nombres, 1900, 335-9, 375-390. 
•'BuU. Amer. Math. Soc, 7, 1901, 350. 
•'Vorlesungen liber Zahlentheorie, I, 1901, 416-428. 
