The square of every prime number can be expressed in a linear form.

Question

While working on the Collatz conjecture, I've found that the square of every prime number $p$ (except 2 and 3)can be written in the form of $12k+1$.

$p^2=12k+1$.$(k\in\mathcal N)$

is this a new discovery ? or have been used earlier somewhere.

Answer 1

All the primes except $2$ and $3$ are $1,5,7 \text { or }11 \bmod 12$ because all the other residue classes have a divisor of $2$ or $3$. If you square each of those you get $1 \bmod 12$. Your observation is correct. As Will Jagy points out, all the primes except $2$ and $3$ are of the form $24k+1$, which is a consequence of the fact that all odd squares are $1 \bmod 8$