In another post, Show that every prime p>3 is either of the form 6n+1 or of the form 6n+5, it was shown that primes are in the form $(6n+1)$ or $(6n+5)$.
How do you determine if a number in the form $(6n+1)$ is NOT prime? Clearly $25, 121, 289, 529,$ etc., are not prime ($5 \times 5, 11 \times 11, 17 \times 17, 23x23$).
In general, numbers in the form $(6n+1)$ are not prime when they have the factors $(6q-1)(6q-1)$; this is easy to see why when you multiply out the quadratic factors.
In another post, 6n+1 and 6n−1 prime format, there is a sieve that possibly could be adapted to show values that would not be prime; however, is there a general proof showing all numbers that are not prime and in the form $(6n+1)$?
P.S. Thanks to everyone who has contributed to this great site.
By comparing the numbers up to 200, I've established some rules which I am sure hold. Numbers of the form $6n+1$ are not prime if they are also of any of these forms: Note $p,q\in\Bbb Z$ $$(6p-1)^2$$ $$(6p+1)^2$$ $$(6p-1)(6q-1)$$ $$(6p+1)(6q+1)$$