Chap. VII] PeIMITIVE RoOTS, EXPONENTS, INDICES. 195 
G. B. Mathews'^^ reproduced art. 81 of Gauss''' and gave a second proof 
by use of Cauchy's^^ congruence X=0 for n = p — 1. 
K. Zsigmondy^^ treated the problem to find all integers K, relatively 
prime to given integers a and 6, such that a''=b'' (mod K) holds for the 
given integral value <T = y, but for no smaller value. For 6 = 1, it is a 
question of the moduli K with respect to which a belongs to the exponent y. 
Set y=Ilqi*, where the q's are distinct primes and qi the greatest. Then 
all the primes K for which a''=b'' (mod K) holds for (T = 7, but for no smaller 
a, coincide with the prime factors of 
y y 
(a^-6^)n(a««'-6««'). . . 
A = 
n(a^/«-6^/«)... 
in which the products extend over the combinations of qi,q2,--- one, two, . . . 
at a time, provided that, if a''=h'' (mod qi) for (J=^ylqi\ but for no smaller 
(T, we do not include among the K's the prime q^, which then occurs in A 
to the first power only. If the prime p is a K and if p^ is the highest power 
of p dividing A, then p* is the highest power of p giving a K. The com- 
posite i^'s are now easily found. If a and 6 are not both numerically equal 
to unity, it is shown that there is at least one prime K except in the following 
cases: 7 = 1, a-6 = l; 7 = 2, 0+6 = ^2" (/x^l); 7 = 3, a = ±2, 6==f1; 
7 = 6, a = ='=2, 6 = ±l. The case h = \ shows that, apart from the corre- 
sponding exceptions, there exists a prime with respect to which the given 
integer aj^^^X belongs to the given exponent 7. As a corollary, every 
arithmetical progression of the type mT+I ()" = 1? 2, . . .) contains an infini- 
tude of primes. 
Zsigmondy^^ considered the function A^(a) obtained from the above A 
by setting 6 = 1. If a is a primitive root of the prime p = l+7, the main 
theorem of the last paper shows that p divides A^(a). Conversely, I+7 is 
a prime if it divides A. Thus, if all the primes of a set of integers possess 
the same primitive root a, any integer p of the set is a prime if and only if 
Ap_i(a) is divisible by p. Hence theorems due to Tchebychef^^ imply 
criteria for primes. For example, a prime 2^"+l has the primitive root 3 
implies that 2^"+l is a prime if and only if it divides 3^ + 1, where k = 2^'' . 
Since ±2 is a primitive root of any prime 2q-\rX such that g- is a prime 
4/c± 1, we infer that, if g is a prime 4/c± 1, then 2g+l is a prime if and only 
if it divides (2^±1)/(2±1). Since 2 is a primitive root of a prime 4A''+1 
such that N is an odd prime, we infer that, if N is an odd prime, 4A^+1 is a 
prime if and only if it divides (2^^-|-l)/5. 
G. F. Bennett^° proved (pp. 196-7) the first theorem of Cauchy,^^ and 
(pp. 199-201) the results of Sancery.^^ If a and a' belong to exponents 
t and t' which contain no prime factor raised to the same power in each, 
then the exponent to which aa' belongs is the 1. c. m. of t and t' (p. 194). 
"Theory of Numbers, 1892, 23-25. 
"Monatshefte Math. Phys., 3, 1892, 265-284. 
'Hhid., 4, 1893, 79-80. 
8»Phil. Trans. R. Sec. London, 184 A, 1893, 189-245. 
