2021/040) Prove that there are infinitely many positive integers n such that n(n+1) can be expressed as a sum of two squares in atleast two different ways.

 Let the number be $z = n(n+1)$

So $z = n^2 + n$

If we choose n to be a square say $m^2$ then we have z already a sum of 2 squares

We have $z = (m^2)^2 + m^2= m^2(m^2+1)$

If we have $m^2$ as sum of 2 squares say $x^2+y^2$ then 

we have

$z = (p^2 + q^2) (m^2 + 1) = p^2m^2 + q^2m^2 + p^2 + q^2$

$= (p^2m^2 + q^2 - 2pqm) + (q^2m^2 +p^2 + 2pqm)$

$= (pm - q)^2 + (qm +p)^2$

This is another way of representation

As (p,q,m) are sides of a Pythagorean triple and there are infinite Pythagorean triples so there are infinite independent values.