192 History of the Theory of Numbers. [Chap, vii 
C. J. D. Hill^^ noted that his tables of indices for the moduU 2" and 5" 
(n^5) give the residues of numbers modulo 10", i. e., the last n digits. 
Using also tables for the moduli 9091 and 9901, as well as a table of loga- 
rithms, we are able to determine the last 22 digits. 
B. M. Goldberg^^ gave the least primitive root of each prime < 10160. 
V. Bouniakowsky^^ proved that 3 is a primitive root of p if p = 24n+5 
and (p — 1)/4 are primes; —3 is a primitive root of p if p = 12n+ll and 
(p — 1)/2 are primes; if p is a primitive root of the prime p = 4n+l, one 
(or both) of p, p—p is a primitive root of p"' and of 2p"'; 5 is a primitive 
root of p = 20?i+3 or 20n4-7 if p and {p — 1)/2 are primes, and of p = 40n + 13 
or 4071-1-37 if p and (p — 1)/4 are primes; 6 is a primitive root of a prime 
24n-|-ll and —6 of 24n-|-23 if (p — 1)/2 is a prime; 10 is a primitive root 
of p = 40n+7, 19, 23, and -10 of p = 40n+3, 27, 39, if (p-l)/2 is a prime; 
10 is a primitive root of a prime 80n-(-73, n>0, or 80n+57, n>l, if 
(p — 1)/8 is a prime. If p = 8an-h2a — 1 or 8an+a— 2 and (p — 1)/4 are 
primes, and if a^-f-1 is not divisible by p, a is a primitive root of p. 
V. A. Lebesgue^^ proved certain theorems due to Jacobi^^ and the 
following theorem which gives a method different from Jacobi's for forming 
a table of indices for a prime modulus p: If a belongs to the exponent n, 
and if 6 is not in the period of a, and if / is the least positive exponent for 
which h^=a\ then x^=a has the root a'6", where ft-\-iu — l=nv; the root 
belongs to the exponent nf if and only if u is prime to /. 
Consider the congruence x*" = a (mod p) , where a belongs to the exponent 
n = (p — l)/n', and m is a divisor of n'. Every root r has a period of mn 
terms if no one of the residues of r, r^,. . ., r*""^ is in the period of a. If all 
the prime divisors of m divide n, the m roots have a period of mn terms; 
but if m has prime divisors g, r, . . . , not dividing n, there are only 
-(^X^)- 
roots having a period of mn terms. The existence of primitive roots follows; 
this is already the case if m = n'. 
Mention is made of companion tables in manuscript giving indices of 
numbers, and numbers corresponding to indices, constructed by J. Ch. 
Dupain in full for p<200, but from 200 to 1500 with reduction to one-half 
in view of ind p — a=ind a=t(p — 1)/2 modulo p — 1. 
L. Kronecker^^ proved the existence of two series of positive integers 
Qj, m, {j=l,. . ., p) such that the least positive residues modulo A:>2 of 
^1 V2*' • • ■ ^p*" give all the (f){k) positive integers <A: and prime to k, if 
ii=0, 1,. . ., mi — 1; i2 = 0, 1,. . ., m2 — 1; etc. [cf. Mertens^^]. 
G. Barillari^"" proved that, if a is prime to h and belongs to the exponent 
"Jour, fur Math., 70, 1869, 282-8; Acta Univ. Lundensis, Lund, 1, 1864 (Math.), No. 6, 18 pp. 
"Rest- und Quotient-Rechnung, Hamburg, 1869, 97-138. 
"BuU. Ac. Sc. St. P6tersbourg, 14, 1869, 375-81. 
»«Compte8 Rendua Paris, 70, 1870, 1243-1251. 
"Monatsber. Ak. BerUn, 1870, 881. Cf. Traub, Archiv Math. Phys., 37, 1861, 278-94. 
•KJiomale di Mat., 9, 1871, 125-135. 
