The theory of numbers, by Robert D. Carmichael ...

84

THEORY OF NUMBERS

We have given this example as an elementary illustration
of the analytical theory of numbers, that is, of that part of, the
theory of numbers in which one employs (as above) the theory
of a continuous variable or some analogous theory in order to
derive properties of sets of integers. This general subject
has been developed in several directions. For a systematic
account of it the reader is referred to Bachmann's Analytische
Zahlentheorie.
~ 45. DIOPHANTINE EQUATIONS
Iff(x, y, z,... ) is a polynomial in the variables x, y, z,.
with integral coefficients, then the equation
f(x, y,... )=o
is called a Diophantine equation when we look at it from the
point of view of determining the integers (or the positive integers) x, y, z,... which satisfy it. Similarly, if we have
several such functions fi(x, y,,... ), in number less than
the number of variables x, y, z,..., then the set of equations
fi(x,y,...)=, i=, 2,..,
is said to be a Diophantine system of equations. Any set of
integers x, y, z,... which satisfies the equation [system]
is said to be a solution of the equation [system].
We may likewise define Diophantine inequalities by replacing the sign of equality above by the sign of inequality. But
little has been done toward developing a theory of Diophantine
inequalities. Even for Diophantine equations the theory is
in a rather fragmentary state.
In the next two sections we shall illustrate the nature of
the ideas and the methods of the theory of Dipohantine equations by' developing some of the results for two important
special cases.