Let be positive integers.

We can consider the case when .

In such a case we require that and .

Another observation is that .

Given two positive odd integers it turns out that either or .

These will be our two cases to consider.

Case 1: We can write and so

Notice that,

Now write, with

We have a situation when product of two relatively prime integers is a square, so each integer is a square.

This gives,

Solving this we get,

Also,

This shows that the parametric description of the equations above give us a complete list for solutions.

Case 2:: We can write and so

Notice that,

Now write, with

This gives,

Solving this we get,

Also,

This shows that the parametric description of the equations above give us a complete list for solutions.

Case 1 and Case 2 can be combined by saying: as a complete list