AoPS - Problem

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Problem

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Tags: number theory, diophantine

parmenides51

01.10.2021 21:52

Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$ . Prove that $a-b$ is a perfect square.

rafaello

01.10.2021 22:56

The given is equivalent to $(a-b)(2a+2b+1)=b^2$ . Note that there are no primes $p$ such that $p\mid a-b$ and $p\mid 2a+2b+1$ , otherwise since then also $p\mid b$ , we get that $p\mid a$ and thus $p\mid 1$ . Hence, $a-b$ and $2a+2b+1$ must be perfect squares.