A number theory problem by Áron Bán-Szabó

Number Theory Level 2

Is it possibe, that $n^2+n+1$ is a perfect square, where $n$ is a positive integer?

Yes, it's possible. No, it's not possible.

1 solution

Áron Bán-Szabó
Aug 19, 2017

Note that $n^2$ and $(n+1)^2$ are two consecutive perfect squares. That means there's no perfect square bigger than $n^2$ , but lower than $(n+1)^2$ . However $n^2 \ < \ n^2+n+1 \ < \ (n+1)^2=n^2+2n+1$

Therefore it is not possible.

A number theory problem by Áron Bán-Szabó

1 solution

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