196 History of the Theory of Numbers. [Chap, vii 
If 2*'^^ is the highest power of 2 dividing a^ — 1, where a is odd, the exponent 
to which a belongs modulo 2^ is 2^~' if X>s, but, if X^s, is 1 if a=l, 2 if 
o^=l, a^l (mod 2^^); the result of Lebesgue'*^ (p. 87) now follows (pp. 
202-6) . In case a is not prime to the modulus, there is an evident theorem 
on the earliest power of a congruent to a higher power (p. 209). If e is a 
given divisor of 0(w), there is determined the number of integers belonging 
to exponent e modulo m [cf. Erlerus^^]. If a, a',. . . belong to the exponents 
t, t',... and if no two of the «' . . . numbers a'a"' . . . {0^r<t,0^r'<t',. . .) 
are congruent modulo m, then a, a', . . . are called independent generators 
of the 4>{m) integers <m and prime to m (p. 195); a particular set of 
generators is given and the most general set is investigated (pp. 220-241) 
[a special problem on abelian groups]. 
J. Perott^^ found a number belonging to an exponent which is the 1. c. m. 
of the exponents to which given numbers belong. If, for a prime modulus p, 
a belongs to an exponent t>l, and b to an exponent which divides t, then b 
is congruent to a power of a (proof by use of Newton's relations between 
the sums of like powers of a, . . . , a' and their elementary symmetric func- 
tions). Hence there exists a primitive root of p. 
M. Frolo . ^- noted that all the quadratic non-residues of a prime modulus 
m are primitive roots of m if m = 2^''4-l, m = 2n+l or 4n + l with n an odd 
prime [Tchebychef^^]. To find primitive roots of m "without any trial," 
separate the m — 1 integers <m into sets of fours a, b, —a, —b, where 
a6=l (mod m): Begin with one such set, say 1, 1, —1, —1. Either a or 
m — a is even; divide the even one by 2 and multiply the corresponding 
=t 6 by 2 ; we get another set of four. Repeat the process. If the resulting 
series of sets contains all m — 1 integers <m, 2 and —2 are primitive roots 
if w = 4/i+l, and one of them is a primitive root if m = 4/i — 1. If the sets 
just obtained do not include all m — 1 integers <m, further theorems are 
proved. 
G. Wertheim^^ gave the least primitive root of each prime p <3000. 
L. Gegenbauer^^" gave two expressions for the sum Sk of those terms of a 
complete set of residues modulo p which belong to the exponent k, and 
evaluated l>Sk/t fit) with t ranging over the divisors of k. 
G. Wertheim^^ proved that any prime 2'*" + l has the primitive root 7. 
If p = 2"g-|-l is a prime and ^ is a prime >2, any quadratic non-residue m 
of p is a primitive root of p if m"" — 1 is not divisible by p. As corollaries, 
we get primes q of certain linear forms for which 2, 5, 7 are primitive roots 
of a prime 2^-1-1 or 4g-f-l; also, 3 is a primitive root of all primes 8g-|-l 
or 16g+l except 41; and cases when 5 or 7 is a primitive root of primes 
8^+1, lQq+1. There is given a table showing the least primitive root of 
each prime between 3000 and 3500. 
"BuU. des Sc. Math., (2), 17, I, 1893, 66-83. 
""BuU. Soc. Math, de France, 21, 1893, 113-128; 22, 1894, 241-5. 
"Acta Mathematica, 17, 1893, 315-20; correction, 22, 1899, 200 (10 for p = 1021). 
8"Denkschr. Ak. Wiss. Wien (Math.), 60, 1893, 48-60. 
"Zeitschrift Math. Naturw. Unterricht, 25, 1894, 81-97. 
