194 History of the Theory of Numbers. [Chap, vii 
J. Perott^^ gave a simple proof that x^ = l (mod p") has p'' roots. Thus 
there exists an integer b belonging to the exponent p""^ modulo p"*. Assum- 
ing the existence of a primitive root of p, we employ a power of it and obtain 
a number a belonging to the exponent p — 1 modulo p". Hence ab is a 
primitive root of p". 
Schwartz^" stated, and Hacken proved, the final theorem of Cauchy.^* 
L. Gegenbauer^^ stated 19 theorems of which a specimen is the follow- 
mg: If p = 8a(8/3+l) + 24/3+5 and (p-l)/4 are primes and if 64a2+48a 
+ 10 is relatively prime to p, then 8a+3 is a primitive root of p. 
G. Wertheim^^ gave the least primitive root of each prime < 1000 and 
companion tables of indices and numbers for primes < 100. He reproduced 
(pp. 125-130) arts. 80-81 of Gauss^ and stated the generalization by 
Stern.i^ 
H. Keferstein''' would obtain all primitive roots of a prime p by excluding 
all residues of powers with exponents dividing p — 1 [Poinsot^]. 
IM. F. Daniels"^ gave a proof like Legendre's^ that there are <f>{n) num- 
bers belonging to the exponent n modulo p, a prime, if n divides p — 1. 
*K. Szily^- discussed the "comparative number" of primitive roots. 
E. Lucas"^ gave the name reduced indicator of n to Cauchy's^^ maximum 
indicator of n, and noted that it is a divisor <4){n) of 0(n) except when 
n = 2, 4, p* or 2p^', where p is an odd prime, and then equals (f>{n). The 
exponent to which a belongs modulo m is called the "gaussien" of a modulo 
m (preface, xv, and p. 440). 
H. Scheffler"'* gave, without reference, the theorem due to' Richelot^'^ and 
the final one by Prouhet.^- To test if a proposed number a is a primitive 
root of a prime p, note whether p is of one of the linear forms of primes for 
which a is a quadratic non-residue, and, if so, raise a to the powders whose 
exponents divide (p — 1)/2. 
L. Contejean^^ noted that the argument in Serret's Algebre, 2, No. 318, 
leads to the following result [for the case a = 10]: If p is an odd prime and 
a belongs to the exponent e = {p — l)/q modulo p, it belongs to the exponent 
p-'^e modulo p" when (a*— l)/p is not di\dsible by p, but to a smaller 
exponent if it is divisible by p [Sancery®^]. 
P. Bachmann^^ proved the existence of a primitive root of a prime p 
by use of the group of the residues 1, . . . , p — 1 under multiplication. 
**Bull. des Sc. Math., 9, I, 1885, 21-24. For k = n — l the theorem is contained imphcitly in a 
posthumous fragment by Gauss, Werke, 2, 266. 
"Mathesis, 6, 1886, 280; 7, 1887, 124-5. 
«8Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 843-5. 
"Elemente der Zahlentheorie, 1887, 116, 375-381. 
'"Mitt. Math. Gesell. Hamburg, 1, 1889, 256. 
^'Lineaire Congruenties, Diss., Amsterdam, 1890, 92-99. 
"Math, ^s termes ^rtesito (Memou^ Hungarian Ac. Sc), 9, 1891, 264; 10, 1892, 19. Magyar 
Tudom. Ak. Ertesitoje (Report of Hungarian Ac. Sc), 2, 1891, 478. 
"Th^orie des nombres, 1891, 429. 
'*Beitrage zur Zahlentheorie, 1891, 135-143. 
"Bull. Soc. Philomathique de Paris, (8), 4, 1891-2, 66-70. 
"Die Elemente der Zahlentheorie, 1892, 89. 
