## Infinite solutions

For discussing Olympiad Level Number Theory problems
aritra barua
Posts: 50
Joined: Sun Dec 11, 2016 2:01 pm

### Infinite solutions

Find with proof every pair of ($a$,$b$) in general form such that $a+b^2$|$a^3+b^3$.

$-a \equiv b^2 (moda+b^2)$.
So, $-a^3 \equiv b^6 (moda+b^2)$.
Which implies that $a+b^2 | b^6-b^3$.
Now fix $b$ and notice that for every divisor $x$ of $b^6-b^3$ there exists a $a$ such that $a+b^2=x$. So we get infinite solutions.