I am trying but still unsuccessfully solving the following question:
For every positive integer $n$, show that any prime divisor of $12n^2 + 1$ is of the form $6k + 1$, where $k$ is an integer.
I would like, if possible, to get a solution that does not refer to Legendre symbol or quadratic reciprocity. That is: I would like to get, if possible, a more elementary solution.
I ask for help.