What is the largest value of $n$ for which $2n + 1$ is a factor of $122 + n^{2}$? Given that $n$ is a natural number, what is its largest value such that  $2n + 1$ is a factor of  $122 + n^{2}$?

Note that $4\cdot(n^2+122) - (2n+1)(2n-1)=489$. Hence if $n^2+122$  is a multiple of $2n+1$, we also need $2n+1\mid 489=2\cdot 244+1$. On the other hand, $n=244$ does indeed lead to $n^2+122 = 122\cdot(2n+1)$, so the largest $n$ is indeed $244$.
