PREFACE. VII 
The properties of the set of all irreducible fractions, arranged in order of 
magnitude, whose numerators are ^ m and denominators are ^ n (called a 
Farey series if m = n), have been discussed by many writers and applied to 
the approximation of numbers, to binary quadratic forms, to the composi- 
tion of linear fractional substitutions, and to geometry (pp. 155-8). 
Some of the properties of periodic decimal fractions are already familiar 
tq the reader in view of his study of arithmetic and the chapter of alge- 
bra dealing with the sum to infinity of a geometric progression. For the 
generalization to periodic fractions to any base h, not necessarily 10, the 
length of the period of the periodic fraction for 1/d, where d is prime to h, 
is the least positive exponent e such that h^ — \ is divisible by d. Hence this 
Chapter VI, which reports upon more than 160 papers, is closely related to 
the following chapter and furnishes a concrete introduction to it. 
The subject of exponents and primitive roots is one of the important 
topics of the theory of numbers. To present the definitions in the customary, 
compact language, we shall need the notion of congruence. If the differ- 
ence of two integers a and 6 is divisible by m, they are called congruent 
modulo m and we write a=& (mod m). For example, 8=2 (mod 6). If 
n'= 1 (mod m), but n*^ 1 (mod m) for 0<s<e, we say that n belongs to the 
exponent e modulo m. For example, 2 and 3 belong to the exponent 4 
modulo 5, while 4 belongs to the exponent 2. In view of Euler's generaliza- 
tion of Fermat's theorem, stated above, e never exceeds 0(m). If n belongs 
to this maximum exponent ^(n) modulo m, n is called a primitive root of m. 
For example, 2 and 3 are primitive roots of 5, while 1 and 4 are not. Lam- 
bert stated in 1769 that there exists a primitive root of any prime p, and 
Euler gave a defective proof in 1773. In 1785 Legendre proved that there 
are exactly 4>{e) numbers belonging modulo p to any exponent e which 
divides p — 1. In 1801 Gauss proved that there exist primitive roots of m 
if and only if m = 2, 4, p* or 2p*, where p is an odd prime. In particular, for 
a primitive root a of a prime modulus p and any integer N not divisible 
by p, there is an exponent ind N, called the index of N by Gauss, such that 
N=a''"^^ (mod p). Indices play a role similar to logarithms, but we re- 
quire two companion tables for each modulus p. The extension to a power 
of prime modulus is immediate. For a general modulus, systems of indices 
were employed by Dirichlet in 1837 and 1863 and by Kronecker in 1870. 
Jacobi's Canon Arithmeticus of 1839 gives companion tables of indices for 
each prime and power of a prime < 1000. Cunningham's Binary Canon of 
1900 gives the residues of the successive powers of 2 when divided by each 
prime or power of a prime < 1000 and companion tables showing the powers 
of 2 whose residues are 1, 2, 3, . . .. In 1846 Arndt proved that, if ^ is a 
primitive root of the odd prime p, g belongs to the exponent p"~"^(p — 1) 
modulo p'* if and only ii G = g^~^ — 1 is divisible by p^, but not by p^'^\ where 
