Chap. VII] PrBIITIVE RoOTS, EXPONENTS, INDICES. 193 
m modulo h, and if h^ is the highest power of h which divides a"* — 1, and if 
n^/i, then 6" divides a* — 1 where e = m6"~\ Further, if 6 is a prime, a 
belongs to the exponent e modulo 6". For a new prime 6', let m', n', h' 
have the corresponding properties. Then the exponent to which a belongs 
modulo B = })%'"' ... is the 1. c. m. L of m6''-\ m'b'"'-''', .... For a = 10, we 
see that L is the length of the period for the irreducible fraction N/B. 
L. Sancery^^ proved that if p is a prime and a<p belongs to the exponent 
6 modulo p, there exists an infinitude of numbers a-\-px = A such that A^—1 
is divisible by p^, but not by p'''^^, where k is any assigned positive integer. 
If A belongs to the exponent 6 modulo p>2, A will belong to the exponent 
6 modulo p" if the highest power of p which divides .A^ — 1 is ^p"; but if it 
be p"'^, A belongs to the exponent dp^ modulo p" [Barillari^°"]. Hence A 
is a primitive root of p" if a primitive root of p and if A^~^ — 1 is not divisible 
by p^, and there are ^j^CpOj primitive roots of p" or 2p\ [Generalization 
of Arndt.^^j 
C. A. Laisant®^ noted that if a belongs to the exponent 3 modulo p, a 
prime, then a + 1 belongs to the exponent 6, and conversely. If a belongs to 
the exponent 6, a+1 will not belong to the exponent 3 unless p = 7, a = 3. 
Hence if p is a prime 6m +1, there are two numbers a, h belonging to the 
exponent 3, and two numbers a + 1, 6+1 belonging to the exponent 6; also, 
a+6 = p — 1. If (p. 399) p+5 is an odd prime and p is even, then pV— 9> 
p^qP = p (mod p+g). 
G. Bella vitis^^'' gave, for each power p'^383 of a prime p, the periodic 
fraction for 1/p' to the base 2 and showed how to deduce the indices of 
numbers for the modulus p\ Let ? = p'~^(p — 1) and let 2 belong to the 
exponent q/r modulo p\ A root b of 6'"= 2 (mod p') is the base of the 
system of indices. 
G. Frattini^^ proved by the theory of roots of unity that, if p is a prime, 
the number of interchanges necessary to pass from 1, 2, . . ., p — 2 to ind 2, 
ind 3, . . . , ind (p — 1) and to 
ind 1— ind 2, ind 2 — ind 3, . . ., ind (p — 2)— ind (p — 1) 
are both even or both odd. 
Fritz Hofmann^^ used rotations of regular polygons to prove theorems 
on the sum of the primitive roots of a prime (Gauss^). 
A. R. Forsyth^^ found the sum of the cth powers of the primitive roots 
of a prime p. The sum is divisible by p if p — 1 contains the square of a 
prime not dividing c or if it contains a prime dividing c but with an exponent 
exceeding by at least 2 its exponent in c. If neither of these conditions is 
satisfied, the result is not so simple. 
"BuU. Soc. Math, de France, 4, 1875-6, 23-29. 
fi^M^m. Soc. So. Phys. et Nat. de Bordeaux, (2), 1, 1876, 400-2. 
"^lAtti Accad. Lincei, Mem. Sc. Fis. Mat., (3), 1, 1876-7, 778-800. 
"Giornale di Mat., 18, 1880, 369-76. 
"Math. Annalen, 20, 1882, 471-86. 
^'Messenger of Math., 13, 1883-4, 180-5. 
