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

ELEMENTARY PROPERTIES OF CONGRUENCES

39

As a corollary of theorems II, III and V we have the following more general result:
VI. If f(x) denotes any polynomial in x with coefficients
which are integers (positive or zero or negative) and if further
a-b mod m, then
f(a) =f(b) mod m.
~ 19. SOLUTIONS OF CONGRUENCES BY TRIAL
Let f(x) be any polynomial in x with coefficients which
are integers (positive or negative or zero). Then if x and c
are any two integers it follows from the last theorem of the
preceding section that
f(x) (-f(x +cm) mod m. (I)
Hence if a is any value of x for which the congruence
f(x)=o mod m (2)
is satisfied, then the congruence is also satisfied for x=a+cm,
where c is any integer whatever. The numbers a+cm are
said to form a solution (or to be a root) of the congruence, c
being a variable integer. Any one of the integers a+cm may
be taken as the representative of the solution. We shall often
speak of one of these numbers as the solution itself.
Among the integers in a solution of the congruence (2)
there is evidently one which is positive and not greater than
m. Hence all solutions of a congruence of the type (2) may
be found by trial, a substitution of each of the numbers I, 2,.. being made for x. It is. clear also that m is the maximum number of solutions which (2) can have whatever be
the function f(x). By means of an example it is easy to show
that this maximum number of solutions is not always possessed
by a congruence; in fact, it is not even necessary that the
congruence have a solution at all.
This is illustrated by the example

x2- 3 o mod 5.