Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 183 
The product of all the primitive roots of a prime p^3 is =1 (mod p) ; 
the sum of the primitive roots of p is =0 if p — 1 is divisible by a square, 
but is =( — 1)" if p — 1 is the product of n distinct primes (arts. 80, 81). 
If p is an odd prime and e is the g. c. d. of ^(p")=p''~^(p — 1) and t, 
then x' = l (mod p") has exactly e incongruent roots. It follows that there 
exist primitive roots of p", i. e., numbers belonging to the exponent 0(p") 
(arts. 85-89). 
For n>2, every odd number belongs modulo 2" to an exponent which 
divides 2""^, so that primitive roots of 2" are lacking; however, a modified 
method of employing indices to the base 5 may be used (arts. 90, 91). 
If w = A"5^.., where A, B,... are distinct primes, and a=0(A"), 
^=4>{B^), . . ., and if ii is the 1. c. m. of a, jS, . . ., then ^'' = 1 (mod m) for z 
prime to m. Now fiKa-^. . . =4>{m) except when m = 2", p" or 2p", where 
p is an odd prime. Thus there exist primitive roots of m only when m = 2, 
4, p"or2p" (art. 92). 
Table I, at the end of Disq. Arith., gives on one page the indices of each 
prime <p for each prime and power of prime modulus < 100. Gauss gave 
no direct table to determine the number corresponding to a given index, 
but indicated (end of art. 316) how his Table III for the conversion of ordi- 
nary into decimal fractions leads to the number having a given index (cf. 
Gauss,i^'i^Ch. VI). 
S. F. Lacroix^ reproduced Gauss' second proof of the existence of primi- 
tive roots of a prune, without a reference. 
L. Poinsot^ argued that the primitive roots of a prime p may be obtained 
from the algebraic expressions for the imaginary (p — l)th roots of unity 
by increasing the numbers under the radical signs by such multiples of p 
that the radicals become integral. The (/)(p — 1) primitive roots of p may 
be obtained by excluding from 1, . . . , p — 1 the residues of the powers whose 
exponents are the distinct prime factors of p — 1; while symmetrical, this 
method is unpractical for large p. 
Fregier^° proved that the 2"th power of any odd number has the remainder 
unity when divided by 2""*"^, if n>0. 
Poinsot^^ developed the first point of his preceding paper. The equa- 
tion for the primitive 18th roots of unity is x^—x^-\-l=0. The roots are 
: = a^^Kl + ^^'=^ (a' = l). 
But \/^= ±4, -¥^=4:, ^-11 = 2 (mod 19). Thus the six primitive 
roots of 19 are x= —4, 2, —9, —5, —6, 3. In general, the algebraic expres- 
sions for the nth roots of unity represent the different integral roots of 
a;" = l (mod p), where p is a prime kn-\-\, after suitable integers are added 
to the numbers under the radical signs. Since unity is the only (integral) 
sCompldment des ^l^mens d'alg^bre, Paris, ed. 3, 1804, 303-7; ed. 4, 1817, 317-321. 
«M6m. Sc. Math, et Phys. de I'Institut de France, 14, 1813-5, 381-392. 
"Annales de Math, (ed., Gergonne), 9, 1818-9, 285-8. 
"M6m. Ac. Sc. de I'Institut de France, 4, 1819-20, 99-183. 
