TITLE: Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.
QUESTION [13 upvotes]: Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.($a^2+b^2$, is same $b^2+a^2$), $n \in \mathbb{N}.$

I have proved the above question which appeared in one of the Math-Olympiad. And I do know the solution. Sharing the question only because the question has a cute solution.

REPLY [5 votes]: Pretty happy with approaches. Considering $n$ as a square gives one  fine way.
Consider $n=m^2=p^2+q^2$
Now, $n(n +1)= (p^2+q^2) (m^2+1)$
$=(pm+q)^2+(qm-p)^2$
Note that they are two distinct ways. 
Thus, for example, $m=5k, p=4k,q=3k$
$n(n +1)= (25k^2)^2+ (5k)^2=(15k^2+4k^2)^2+(20k^2-3k^2)^2$
And we know that there are infinite numbers of the form $n=p^2+q^2$ (Pythagorean Triplets)