Pythagorean triples can be generated by the expressions
m^2 - n^2, 2mn, m^2 + n^2
since
(m^2 - n^2)^2 + (2mn)^2
= m^4 - 2(m^2)(n^2) + n^4 + 4(m^2)(n^2) = m^4 + 2(m^2)(n^2) + n^4
= (m^2 + n^2)^2.
You can choose any integer values for m and n, providing that n > 0 and m > n.
For example, m = 2 and n = 1 gets you the well known 3, 4, 5 case.
m = 5, n = 2 produces 21, 20, 29, (check 21^2 + 20 ^2 = 441 + 400 = 841 = 29^2), etc.
An infinite number of possibilities.
- Bertie