Hi, as I was messing around with primes I came to the realization that for every prime larger than 3, the following two things are (independently) true:
(p > 3)
(p±1) is divisible by 4.
(p±1) is divisible by 3. (And also 6)
It is important to highlight that ± means or - here, since at least one is true. And independently means that if (p 1) is divisible by 4, that does not relate to the sign in (p±1) mod 3 = 0.
I tried to find this on the internet, but I couldnt, so I assume this isnt too significant.
While this can give us an idea of the distribution of primes, I dont think that anything can be done with this information. Here comes my first question, can something be done with this information?

Secondly, as I was checking this property for the first n numbers, I found that if (p 1) isnt divisible by either 3 or 4, then (p-1) must be divisible by both 3 and 4 and vice versa, if (p-1) isnt divisible by either 3 or 4, then (p 1) must be divisible by both 3 and 4.
Which is pretty straightforward in hindsight. Again, I didnt found anything about this either. Im not sure if it's because there isnt anything about it or because I didnt use the proper name for it, but my question remains the same: can we do anything with this information?

Thanks for the answers.