Sophie Germain Primes: both p and 2p+1 are prime
This is an example of how prime number patterns can be deciphered using the Prime Spiral Sieve as an analytical tool; in this case Sophie Germain Primes.
We start the analysis with the knowledge that all prime numbers (p), with the exception of 2, 3 and 5 are p ≡ 1, 7, 11, 13, 17, 19, 23 or 29 (modulo 30), the set of which we will dub mod30pr, for modulo 30 prime roots. It follows that both p and 2p+1 must be mod30pr. Doing a mod 30 analysis of all the possible combinations of this pattern, only three give results for p and 2p+1 = mod30pr: 11, 23 and 29 (all three of which are themselves Sophie Germain Primes). Thus p for all primes falling into this pattern must be p ≡ 11, 23 or 29 (mod 30). Therefore, if you're seeking these particular primes you can drastically narrow your search.
For example, starting with the mod30pr,11 series (11, 41, 71, 101, 131, 161, 191 ...n) we get:
11 and 2∗11+1 compute to the primes 11 and 23 (making 11 a Sophie Germain Prime)
71 and 2∗71+1 compute to 71, a prime, and 143 (the first composite number in this sequence, thus disqualifying 71 as a Sophie Germain Prime)
... and then on to 101, 131(SGP), 161, 191(SGP), 221, 251(SGP), 281(SGP), 311, etc... n
Moving on to mod30pr,23 we get:
23(SGP), 53(SGP), 83(SGP), 113(SGP), 143, 173(SGP), 203, 233(SGP), etc.
Then, lastly, we move on to mod30pr,29:
29(SGP), 59(SGP), 89(SGP), 109, 149, 179(SGP), 209, 239(SGP), 269, 299, etc... n
As you can see, it's a hit-and-miss proposition; upon factorization, the underlying pattern–though its form persists as a repeating potential to be filled with primes to infinity–gets broken with increasing frequency as we move down the number line. This statement can be applied to virtually every repeating pattern involving prime numbers.