Writing It Up · Superchallenge problem

$n^2 + n + 1$ is always odd, and odd numbers are never perfect squares

$n^2 < n^2 + n + 1 < (n + 1)^2$, and no square lies strictly between consecutive squares

$n^2 + n + 1 \equiv 3 \pmod 4$ for every $n$, and squares are $0$ or $1 \pmod 4$

Checking $n = 1, 2, \ldots, 10$ gives $3, 7, 13, 21, 31, 43, 57, 73, 91, 111$, none a square