mluttgens <email@example.com> writes: <snip> > Thank you! You are of course right. > > But my aim was to show that a sum s? = a + b of two uneven numbers, at > least one of them not being a prime, could easily be transformed into > a sum of two primes, simply by adding and subtracting some even number > from its terms: > > The chosen example was: > > s? = 13 + 15 = (13-8) + (15+8) = 5+23 > = (13-2) + (15+2) = 11+17 > > It has been claimed that such transformation could sometimes not be > possible. > I am wondering about which terms a and b should be chosen to justify > that claim. > Till now, I did not find a clue in the litterature, but you have > perhaps a reference?
Your transformation is possible if GC it true and false otherwise. Every counter-example to GC (of which none are known, of course) would be an example of what you seek with s = 1 + (s-1). Computers have checked GC up to about 10^18, but since almost everyone thinks GC is true, why would you go searching for a counter-example?
Every reference in the literature about GC is a reference that will help you in your quest, because your statement about transforming non-prime sums into prime sums is exactly the same as GC.