I have noticed that when adding positive integers to prime numbers, the result may or may not yield another prime number. There does not seem to be a pattern as to when an addition yields another prime. However, there is a set of numbers that does not appear to ever yield another prime when added to a prime.
I do not know how large this set is, and have so far not found any particular pattern within this set, but for numbers from 1 to 360 there are 109 numbers that, when added to any prime, do not yield another prime. I have verified this for the first 500,000 primes.
From this set, 50 are primes themselves and 59 are not.
Also, when mapped on a circle, 80% of these 109 numbers appear 180 degrees opposite of each other.
Has anyone else ever played around with this?