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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.

Deltab
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Mike215
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  • Possibly useful: https://math.stackexchange.com/questions/2741007/arithmetic-sequence-where-every-term-is-prime – MJD May 20 '18 at 14:45

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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 $p,q\in\Bbb Z$ $$(6p-1)^2$$ $$(6p+1)^2$$ $$(6p-1)(6q-1)$$ $$(6p+1)(6q+1)$$

Rhys Hughes
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  • To prove that this holds in general, it's sufficient to note that every odd number that is not divisible by $3$ (i.e., of the form $6p \pm 1$ for $p > 0$) is the product of odd primes greater than $3$; that every odd prime greater than $3$ is not divisible by $3$, and hence is of the form $(6q \pm 1)$ and, finally, that the multiplying two numbers in that form gives a product in the same form. For instance, $(6p+1)(6q+1) = 6(6pq+p+q) + 1$. – Fabio Somenzi May 20 '18 at 15:09
  • Thx for the reply. 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. Are there any other algebraic forms that would show values of 6n+1 that are not prime? – Mike215 May 26 '18 at 15:38
  • Wow fabio , it is an amazing analysis – Aryaman Sep 24 '21 at 08:40