Suppose we want to find all the integers satisfying
is a difference of squares and factorises as
and 7 factorises as
or
so
or
oror
We may equate factors on either side to give either
and
or
and
The first of these gives the simultaneous equations
The solution is
and
The second of these gives the simultaneous equation
The solution is
The first of these gives the simultaneous equations
The solution is
and
The first of these gives the simultaneous equations
The solution isand
If
and
is not a prime number, a great many solutions exist since m+n and m-n must be matched with each factor in turn, either both negative or positive.