186 
History of the Theory of Numbers. 
[Chap. VII 
To find a primitive root g of p, select any convenient integer a and form 
the residues of a, or, a^,. . . [as by Crelle^®]. Let n be the exponent to 
which a belongs. Set nn' = p — \. If n<p — 1, select an integer h not in 
the period a, . . . , a". The residue of 6" is in this period of a. If y is the 
least power of 6 whose residue is in the period of a, then / divides n', say 
w'=i/' (P- xxiii). Since a=g'^', h^=a\ we have 
y^gf^''^g"''^''"'\ h = g^'^'+'^^ (mod p), 
for some value 0, 1, . . . , /—I of k. But A: must be chosen so that i+nk is 
prime to /. For, if i-\-nk = du, where d is a divisor of /, we would have 
5^'' = a". The nf residues of a'b' (r = 0,. . ., n-1; s = 0,. . ., /-I) are dis- 
tinct ; their indices to base g are/', 2f', . . . , nff in some order and are known. 
If nf'<p — l, we employ an integer not in the set a^b' and proceed similarly. 
Ultimately we obtain a primitive root and at the same time the index of 
everj" number. This method was used for the primes between 200 and 1000. 
For primes < 200, the tables by Ostrogradsky^^ w^ere reprinted with the 
same errors (noted at the end of the Canon). 
Jacobi proved that, if n is an odd prime, any primitive root of n^ is a 
primitive root of any higher power of n (p. xxxv). 
For the modulus 2", 4^iu^9, the final tables give the index / of any 
positive odd number to base 3, where 
(_l)(Ar-l)W-3)/8^ = 37 (jjjQ^ 2"). 
Robert ]\Iurphy-^ stated the empirical theorem that every prime 
anr+p has a as a primitive root if p>a/2, p is a prime <a, and if a is a 
primitive root of p. For example, a prime 10nr-{-7 has 10 as a primitive 
root. 
H. G. Erlerus'^ considered two odd primes p and p' and a number m 
such that m=a (mod p), m = a' (mod p'). Let a belong to the exponent 
e modulo p, and a' to the exponent e' modulo p\ If 8 is the g. c. d. of 
e and e', then m belongs to the exponent ee'/8 modulo pp'. He discussed 
at length the number of integers belonging to the exponent n for a com- 
posite modulus. 
A. Cauchy^^ called the least positive integer i for which m' = 1 (mod n) 
the indicator relative (or corresponding) to the base m and modulus n, 
which are assumed relatively prime. If the base m is constant, and ii, 12 
are the indicators corresponding to moduli nj, 112, and if n = nin2 is prime 
to 772, then the 1. c. m. of I'l and {2 is the indicator corresponding to modulus 
n. If the modulus n is constant, and ii, io are the indicators corresponding 
to bases Wi, ^2, and if I'l, 1*2 are relatively prime, then 1*112 is the indicator 
corresponding to the base 7^17^2. 
Let I'l, io be the indicators corresponding to the bases mi, 7722 and same 
modulus n. The g. c. d. 0; of I'l, 2*2 can be expressed (often in several ways) 
as a product uv such that ii/u, io/v are relatively prime. For, if co = a/3. . . , 
"Phil. Mag., (3), 19, 1841, 369. 
"Elementa Doctrinse Numerorum, Diss., Halis, 1841, 18-43. 
»«Comptes Rendus Paris, 12, 1841, 824-845; Oeuvres, (1), 6, 124-146. 
