Suppose we want to find all the integers satisfying

is a difference of squares and factorises asand 7 factorises asorsoorororWe may equate factors on either side to give either

andorand

The first of these gives the simultaneous equations

The solution isand

The second of these gives the simultaneous equation

The solution is

The first of these gives the simultaneous equations

The solution isand

The first of these gives the simultaneous equations

The solution isand

Ifandis 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.