As I was working on my homework last night, I found an interesting pattern with 1mod12. If you look at a number *a* such that $a = 12k+1$ with $k \in \mathbb{Z}, k>0$ then either *a* is prime, or $a=pq$, where *p* and *q* are both prime and $p \equiv q \mod{12}$.

For example:

$13 \equiv 1 \mod{12}$, 13 is prime

$25 \equiv 1 \mod{12}$, $25=5*5, 5 \equiv 5 \mod{12}$

$37 \equiv 1 \mod{12}$, 37 is prime

$49 \equiv 1 \mod{12}$, $49=7*7, 7 \equiv 7 \mod{12}$

$61 \equiv 1 \mod{12}$, 61 is prime

$85 \equiv 1 \mod{12}$, $85=17*5, 17 \equiv 5 \mod{12}$

$109 \equiv 1 \mod{12}$, 109 is prime

$121 \equiv 1 \mod{12}$, $121=11*11, 11 \equiv 11 \mod{12}$

$133 \equiv 1 \mod{12}$, $133=19*7, 19 \equiv 7 \mod{12}$

$157 \equiv 1 \mod{12}$, 157 is prime

$169 \equiv 1 \mod{12}$, $169=13*13, 13 \equiv 13 \mod{12}$

I have no idea if what this means (if it means anything at all), but I haven't found anything through Google on it yet. I also haven't tried to prove it yet, so it might not even hold true for all *k*, but I still thought it was worth posting.