[Zmetta]

6x±1: 5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,55,59,61,65,67,71,73,77,79,83,85,89,91,95,97,

101,103,107,109,113,115,119,121,125,127,131,133,137,139,143,145,149,151,155,157,161,163,167,169,173,175,179,181,185,187,191,193,197,199,203,205,209,211

Is this just a poor sieve for odd-number pairs or is there something more going on within the factors of 6x±1?

[Vvector]

It just eliminates the multiples of 2 and 3

6x ± 0: divisible by 2,3

6x ± 1: not divisible by 2,3

6x ± 2: divisible by 2

6x ± 3: divisible by 3

6x ± 4: divisible by 2

6x ± 5: modulus equivalent to 6x ± 1

[hinkley]

I think I watched a good video from Numberphile and/or Matt Parker on this but I can't seem to find it now. IIRC it was used as an alternative proof for Fermat's last theorem.

This explains the 6n situation pretty concisely though:

https://reflectivemaths.wordpress.com/2011/07/22/proof-prime...

[ladberg]

I think it's just a sieve that removes multiples of 2 and 3, leaving false positives that are multiples of 5, 7, 11, etc.

[caf]

Right, which is why as the numbers get larger and the primes get more sparse, 30 eventually takes over from 6 as the sieve (2x3x5).