Find all solutions in positive integers of each Diophantine equation below.
a)
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