Need good discussion for
- $12|(p + p+2)$, where $p,p+2$ are primes and $> 3$. Why $12$ divides the sum of twin primes?
- $a, ar, ar^2, \ldots $ is a Geometric series. I would like to place $a = 1$ and $r =2$. Then this series becomes $1, 2, 4, \ldots$ Now, $1 + 2 + 4 = 7$ (prime); $1 + 2 + 4 + 8 = 15$ (co-prime);$1 + 2 + 4 + 8 + 16 = 31$ (prime); $1 + 2 + 4 + 8 + 16 + 32 = 63$ (co-prime);$1 + 2 + 4 + 8 +16 + 32 + 64 = 127$ (prime) and so on...
Why this Geometric series is acting like this? I have checked for only few values. So if there is anything wrong, explain. If it is true. Why it is true for in this series.
Also, I would like to know, is there any such kind of Geometric series, which gives us only primes or primes and co-primes alternatively?