Interesting property observed in the prime numbers (Q2721764)

The author considers finite subsets of the prime numbers and how they perform in Goldbach's conjecture. Given a set \(S\) of positive integers, a certain even number greater than 2 is a Goldbach number with respect to \(S\) if it is the sum of two (possibly equal) numbers from \(S\). NEWLINENEWLINENEWLINEHe finds that the set sp(2) (primes that are distances two from another prime) has some remarkable properties. All the even numbers greater than 4208 and less than 360994 are Goldbach numbers for sp(2); it only misses 34 even numbers greater than 2 out of 180496 even numbers investigated. The first is 4 and the other 33 non-Goldbach numbers for sp(2) are all in clusters of three numbers, distances two apart \((94,96,98), (400,402,404) \dots\;\).