202 History of the Theory of Numbers. [Chap, vii 
R. D. CarmichaeP^® called a number a primitive X-root modulo n if it 
belongs to the exponent X(?i), defined in Ch. Ill, Lucas. ""^ The existence 
of primitive X-roots g is proved. The product of those powers of g which 
are prhnitive X-roots is = 1 (mod n) if X(n) >2. A method is given to solve 
X(x) =a, and the solutions tabulated for a ^24. 
C. Posse^-^ noted that in Wertheim's^'^^ table, the primitive root 14 
of 2161 should be replaced by 23, while 10 is not a primitive root of 3851. 
E. Maillet^^^ described the manuscript table by Chabanel, deposited in 
the library of the University of Paris, giving the indices for primes under 
10000 and data to determine the number having a given index. 
F. Schuh^^^ showed how to form the congruence for the primitive roots 
of a prime and gave two further proofs of the existence of primitive roots. 
He treated binomial congruences, quadratic residues and made applica- 
tions to periodic fractions to any base. For any modulus n, he found the 
least m for which x"' = 1 (mod n) holds for every x prime to n, and derived 
the solutions ?i of 4>{n) =m, i. e., n's having primitive roots. 
F. Schuh^^^ discussed the solution of a;' = 1 (mod p") with the least com- 
putation. If X belongs to the exponent q modulo n, the powers of x give 
a cycle of 0(g) numbers each with the "period" q. The numbers prime to 
n and having the period q may form several such cycles — more than one if 
n has no primitive root and q is the maximum period. If n = 2" (a>2), then 
g = 2* (s^a — 2) and the number of cycles is 1, 3 or 2 according as s = 0, s = 1 
or s>l. In the last case, the cj^cles are formed by 2''~^(2fc+l) =f1. 
When q is even, x is said to be of the first or second kind according as 
x'''^= — 1 (mod n) or not. If the numbers of a cycle are of the second kind, 
we get a new cycle of the second kind by changing the signs of the numbers 
of the first cycle. While for moduli n having primitive roots there exist no 
numbers of the second kind, when n has no primitive roots and g is a possible 
even period, there exist at least two cycles of the second kind and of period 
q. Finally, there is given a table showing the number of cycles of each 
kind for moduli ^ 150. 
M. Kraitchik^^^ gave a table showing for each prime p< 10000 a primi- 
tive root of p and the least solutions of 2""=!, 10"= 1 (mod p). 
*J. Schumacher^^*^ discussed indices. 
L. von Schrutka^^^ noted that, if g, r, . . . are the distinct primes dividing 
p — l, where p is a prime, all non-primitive roots of p satisfy 
(a;V-l)(xV_i) . . .=0 (mod p). 
""Bull. Amer. Math. Soc, 16, 1909-10, 232-7. Also, Theory of Numbers, pp. 71-4. 
i^iActa Math., 33, 1910, 405-6. 
i=»L'interm6diaire des math., 17, 1910, 19-20. 
i23Supplement de Vriend derWiskunde, Culemborg, 22, 1910, 34-114, 166-199, 252-9; 25, 1913, 
33-59, 143-159, 228-259. 
"*Ibid., 23, 1911, 39-70, 130-159, 230-247. 
'"Sphinx-Oedipe, May, 1911, Num^ro Special, pp. 1-10; errata listed p. 122 by Cunningham and 
Woodall. Extension to 25000, 1912, 25-9, 39-42, 52-5; errata, 93-4, by Cunningham. 
"'Blatter Gymnasiaj-Schulwesen, Miinchen, 47, 1911, 217-9, 
"^Monatshefte Math. Phys., 22, 1911, 177-186. 
