182 History of the Theory of Numbers. [Chap, vii 
A. M. Legendre^ started with Lagrange's' result that, if p is a prime 
and n is a divisor of p — 1, 
(1) a:" = l(modp) 
has n incongruent integral roots. Let n = v^v'^ . . ., where v, v',. . . are dis- 
tinct primes. A root a of (1) belongs* to the exponent n if no one of 
a"/', a"^', . . . is congruent to unity modulo p. For, if a* = l, 0<d<n, let 
a be the g. c. d. of 6, n, so that a = ny—dz for integers y, z; then 
contrary to hypothesis. Next, of the n roots of (1), n/v satisfy x''^'' = l 
(mod p), and n{l — l/v) do not. Likewise, n{l — l/v') do not satisfy 
2.n/«'' = 1 . qIq i^ jg gaj(j ^Q follow that there are 
*(„)=„ (i-i)(i-i) 
numbers belonging to the exponent n modulo p. If 
^•^^1, ^-^-^Kmodp), 
/3 belongs to the exponent v"". If j8' belongs to the exponent v'^, etc., the 
product /3]S' ... is stated to belong to the exponent n. 
C. F. Gauss^ gave two proofs of the existence of primitive roots of a 
prime p. If d is a divisor of p — 1, and a'' is the lowest power of a congruent 
to unity modulo p, a is said to belong to the exponent d modulo p. Let 
ypid) of the integers 1, 2, . . . , p — 1 belong to the exponent d, a given divisor 
of p — 1 . Gauss showed that i/' (c?) = or (d) , 2 1/' (c^) = p — 1 = 2 <^ (d) , whence 
i^(d) —(t>{d). In his second proof, Gauss set p — 1 =a"6'^. . ., where a, 5, . . . 
are distinct primes, proved the existence of numbers A, B,. . . belonging 
to the respective exponents a", h^,. . ., and showed that AB . . . belongs to 
the exponent p — 1 and hence is a primitive root of p. 
Let a be a primitive root of p, h any integer not divisible by p, and e 
the integer, uniquely determined modulo p — 1, for w^hich o* = 6 (mod p). 
Gauss (arts. 57-59) called e the index of h for the modulus p relative to the 
base a, and wrote e = ind h. Thus 
a'^^^'^b (mod p), ind 66'=ind 6+ind h' (mod p-1). 
Gauss (arts. 69-72) discussed the relations between indices for different 
bases and the choice of the most convenient base. 
In articles 73-74, he gave a convenient tentative method for finding a 
primitive root of p. Form the period of 2 (the distinct least positive resi- 
dues of the successive powers of 2); if 2 belongs to an exponent ^<p — 1, 
select a number 6<p not in the period of 2, and form the period of 6; etc. 
If a belongs to the exponent t modulo p, the product of the terms in the 
period of a is = ( — 1)'+^ (mod p), while the sum of the terms is =0 unless 
a=l (arts. 75, 79). 
•M6m. Ac. R. Sc, Paris, 1785, 471-3. Thdorie des nombres, 1798, 413-4; ed. 3, 1830, 
Nos. 341-2; German transl. by Maser, 2, pp. 17-18. 
*This term was introduced later by Gauss.'' 
^Disquisitiones Arith., 1801, arts. 52-55. 
