Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 189 
E. Desmarest" devoted the last 86 pages of his book to primitive roots; 
the 70 pages claimed to be new might well have been reduced to five by 
the omission of trivial matters and the use of standard notations. To find 
(pp. 267-8) a primitive root of the prime P = 6g+l, where q is an odd prime, 
seek an odd solution of ^^^+3 = (mod P) and set w = 2/2 — 1; then R^=—l 
and R belongs to the exponent 6; thus we know the solutions of x^ = l\ 
let a be any integer prime to P and not such a solution; if a^=±l, then 
=ta belongs to the exponent q, and ±ai2 is a primitive root of P; but, if 
a^« 7^ 1 , then a^^= =f 1 (mod P) , and =*= a is a primitive root of P. If P = 8Q + 1 
and Q are primes, then P=5 (mod 12) and 3 is a quadratic non-residue and 
hence a primitive root of P. 
Let P be a prime of the form 5q^2. Then u^= 5 (mod P) is not solvable. 
Thus, if a is a primitive root of P, 5 = a% where e is odd. Thus if e is prime 
to P — 1, 5 is a primitive root of P. It is recommended that 5 be the first 
number used in seeking by trial a primitive root. And yet he announced 
the theorem (p. 283) that 5 is in general a primitive root. If P is a prime 
5g±2 also of the form 2"Q+1, where Q is an odd prime including 1, then 
(pp. 284-6) 5 is a primitive root of P provided P is not a factor of 5^ —1. 
He gave the factors of the latter and of 10^" — 1 for n = 1, . . . , 5. ' 
Results, corresponding to those just quoted for 5, are stated for p = 7, — 7, 
10, 17. What is really given is a Hst of the linear forms of the primes P 
for which p is a quadratic non-residue. If, in addition, P = 2''Q + 1, where 
Q is an odd prime, then p is a primitive root, provided p^^^^l (mbd P). 
The last condition is ignored in his statement of his results and again in his 
collection (pp. 297-8) of "principles which give primitive roots" entered in 
his table (pp. 298-300) giving a primitive root of each prime < 10000. 
V. A. Lebesgue^^ proved that, if a and p = 2'a+l are primes, any quad- 
ratic non-residue x of p is a primitive root of p if 
a;2*-'+1^0(modp). 
J. P. Kulik^^ gave for each prime p between 103 and 353 the indices and 
all the primitive roots of p. His manuscript extended to 1000. There is 
an initial table giving the least primitive root of the primes from 103 to 1009. 
G. 01tramare^° called x a root of order or index m of a prime piix belongs 
to the exponent {p — l)/m modulo p. Let Xm{x) = (mod p) be the con- 
gruence whose roots are exclusively the roots of order moi p. By changing 
X to x^^"", we obtain Xmn=<l>{^) ^0. li rii, n2, . . . , n are the divisors > 1 of w. 
Am — ■ 
Y Y 
'^Th^orie des nombres. Traits de I'analyse ind6terminee du second degr6 k deux inconnues 
suivi de I'application de cette analyse k la recherche des racines primitives avec une table 
de ces racines pour tous les nombres premiers compris entre 1 et 10000, Paris, 1852, 
308 pp. For errata, see Cunningham, Mess. Math., 33, 1903, 145. 
58Nouv. Ann. Math., 11, 1852, 422-4. 
'9 Jour, fur Math., 45, 1853, 55-81. 
"7Wd., 303-9. 
