Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 191 
H. J. S. Smith^^ stated that some of Oltramare's^^ general results are 
erroneous at least in expression, and gave a simple proof that 0^"^= 1 (mod p**) 
has exactly d roots if d divides 0(p"). 
V. A. Lebesgue^^ proved that, if p is an odd prime and a, b belong to 
exponents a, (3, there exist numbers belonging to the 1. c. m. m of a, (3, as 
exponent. Hence if neither a nor /3 is a multiple of the other, w exceeds 
a and /3. If d<p — l is the greatest of the exponents to which 1, . . ., p — 1 
belong, the latter do not all belong to exponents dividing d, since otherwise 
they would give more than d roots of x'^=l (mod p). Hence there exist 
primitive roots of p. If a is odd, ±l+2°a belongs to the exponent 2™~" 
modulo 2"" (p. 87). If h belongs to the exponent k modulo p, a prime, then 
h+Pz belongs modulo p" to an exponent which divides A;p"~^ (p. 101). If 
/ is a primitive root of p, and f^~^ — l=pz, then / is a primitive root of p™ 
if and only if z is not divisible by p (p. 102). 
G. L. Dirichlet^^ proved the last theorem and explained his^^ system of 
indices for a composite modulus. 
V. A. Lebesgue^° published tables, constructed by J. Hoiiel,^^ of indices 
and corresponding numbers for each prime and power of prime modulus 
< 200, which differ from Jacobi's^^ only in the choice of the least primitive 
root. There is an auxiliary table of the indices of x\ for prime moduli 
<200. 
V. A. Lebesgue^^ stated that, if g'<p is a primitive root of the prime p 
and if g'=g^~^ (mod p), then g' is a primitive root of p; at least one of g and 
g' is a primitive root of p" for n arbitrary. 
V. Bouniakowsky^^ proved in a new way the theorems of Tchebychef^* 
that 2 is a primitive root of p = 8n+3 if p and 4n+l are primes, and of 
p = 4nH-l if p and n are primes. He gave a method to find the exponent 
to which 2 or 10 belongs modulo p. 
A. Cayley^^ gave a specimen table showing the indices a, j3,. . . for every 
number M = a"6^. . .(modiV), where ilf<iV and prime to iV, for iV = l,. . ., 50. 
There is no apparent way of forming another single table for all A^'s analo- 
gous to Jacobi's tables (one for each N) of numbers corresponding to given 
indices. 
F. W. A. Heime^^ gave the least primitive root of each prime < 1000. 
His other results are not new. A secondary root of a prime p is one belong- 
ing to an exponent < p — 1 modulo p. 
"British Assoc. Report, 1859, 228; 1860, 120, §73; Coll. Math. Papers, 1, 50, 158 (Report on 
theory of numbers). 
**Introd. th^orie des nombres, 1862, 94-96. 
"Zahlentheorie, §§128-131, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. 
soM^m. soc. sc. phys. et nat. de Bordeaux, 3, cah. 2, 1864-5, 231-274. 
"Formiiles et tables numer., Paris, 1866. For moduli ^ 347. 
^Comptes Rendus Paris, 64, 1867, 1268-9. 
"Bull. Ac. Sc. St. Petersbourg, 11, 1867, 97-123. 
"Quart. Jour. Math., 9, 1868, 95-96. 
"Untersuchungen, besonders in Bezug auf relative Primzahlen, primitive u. secundare Wurzeln, 
quadratische Reste u. Nichtreste; nebst Berechnung der kleinsten primitiven Wurzeln 
vorf alien Primzahlen zwischen 1 und 1000. BerUn, 1868; ed. 2, 1869. 
