So, coming off of this question, I know how to find out what the remainder is, so after figuring whether the remainder is $1$ or $5$, would I just plug in $p = 6q + (1\ \text{or}\ 5)$ into $p^2+1$?
As in, making it $(6q+1)^2+1$ or $(6q+5)^2+1$?
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Try expanding the expressions $(6q + 1)^2 + 1$ and $(6q + 5)^2 + 1$, then factor once you have simplified each expression. Are the expressions you obtain divisible by a number larger than $1$? – N. F. Taussig Oct 10 '14 at 21:10
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hint: simply prove that if $p \geq 3$ is prime, then $p^2+1$ is composite and this result follows. xp – genisage Oct 10 '14 at 21:36
