Let $x$ and $y$ be two positive integers such that $x+y< 537$. Under these conditions, find the sum of all distinct values of y such that $x^2+y^2=x^3$.
Unsure as to how to go about this question. I know for a fact that the answer is 812, but am not sure how to do it.
We have that $$x^2 + y^2 = x^3$$ $$\Rightarrow y^2 = x^3- x^2 = x^2 (x-1) $$ $$\Rightarrow y = x\sqrt {x-1} $$
It is obvious that for $y$ to belong in $\mathbb Z^{+} $, we should have that $x-1 = k^2$ for $k\in \mathbb N $. Also, we have the condition that $x+y <537$. The permissible values of $k$ for which this is satisfied belongs to $[1,7] \in \mathbb N $. This can be easily checked.
Now summing up the corresponding $y$ terms gives us the value as $\boxed {812} $ confirming your thought. Hope it helps.