Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 187 
where a, j8, . . . are powers of distinct primes, use a as a factor in forming u 
in case a is prime to ii/a, but as a factor of y in case a is prime to zVa, and 
as a factor of either u or v indifferently in case a is prime to both ii/a and 
12/0. Since ii/u and i2/v are relatively prime indicators corresponding to 
bases mi" and m2^ it follows from the preceding theorem that the indicator 
corresponding to base mi"-W2'' and modulus n is 
ii ^2 iii2 1 r • • 
= — = 1. c. m. of ii, i2. 
U V £0 
Hence, given several bases mi, m2, . . . and a single modulus n, we can 
find a new base relative to which the indicator is the 1. c. m. of the indicators 
corresponding to mi, m2, .... If the latter bases include all the integers <n 
and prime to n, the corresponding indicators give all indicators which can 
correspond to modulus n, so that all of them divide a certain maximum 
indicator I. Then for every integer m relatively prime to n,m^ = l (mod n) . 
If n = v°', where v is an odd prime, or if n = 2 or 4, l=^{n). If n = 2^ k>2, 
I=(f}{n)/2. If Ij is the maximum indicator corresponding to a power Uj 
of a prime, and if n = llnj, then I is the 1. c. m. of /i, /2, • • •• The equation 
mx — ny = l has the solution x = 7n^~^ (mod n). 
Cauchy^^ republished the preceding paper, but with an extension from the 
limit n = 100 to the limit n = 1000 for his table of the maximum indicator I. 
C. F. Arndt^^ gave (without reference) Gauss' second proof of the exist- 
ence of a primitive root of an odd prime p, and proved the existence of the 
<^(p") primitive roots of p'* or 2p'', and that there are no primitive roots for 
moduli other than these and 4. If i is a divisor of 2""^, n>2, exactly t 
numbers belong to the exponent t modulo 2'' (p. 18). If, for a modulus 
p", 2p", a belongs to the exponent t, then a-a^ . . .a' is congruent to ( — 1)'+^ 
(pp. 26-27), while the product of the numbers belonging to the exponent t 
is congruent to +1 if ^?^ 2 (pp. 37-38). He proved also Stern's^^ theorem 
on the sum of these numbers. He gave the same two theorems also in a 
later paper. ^^ 
L. Poinsot^" used the method of Legendre^ to prove the existence of 
4>{n) integers belonging to the exponent n, sl divisor of p — 1, where p is a 
prime. He gave (pp. 71-75) essentially Gauss' first proof, and gave his 
own^ method of finding primitive roots of a prime. The existence of 
primitive roots of p", 2p", 4, but of no further moduli, is established by use 
of the number of roots of binomial congruences (pp. 87-101). 
C. F. Arndt^^ noted that if a belongs to an even exponent t modulo 2", 
then ±a, ±a^, ..., ±a'~^ give the t incongruent numbers belonging to 
the exponent t, and are congruent to A; • 2"" =f 1 (A; = 1 , 3, 5, . . . ) . The product 
of the numbers belonging to the exponent t modulo 2", n>2, is = +1. 
• "Exercices d' Analyse et de Phys. Math., 2, 1841, 1-40; Oeuvres, (2), 12. 
"Archiv Math. Phys., 2, 1842, 9, 15-16. 
"Jour, flir Math., 31, 1846, 326-8. 
3«Jour. de Math^matiques, (1), 10, 1845, 65-70, 72. 
"Archiv Math. Phys., 6, 1845, 395, 399. 
