On 24 juil, 16:52, Ben Bacarisse <ben.use...@bsb.me.uk> wrote: > mluttgens <lutt...@gmail.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. > > -- > Ben. > >
I am also almost certain that GC is true, but its validity is not proved. In order to demonstrate that it is false, one could show that a sum of two uneven but not prime numbers cannot be transformed into a sum of primes by adding and subtracting some even number to/from its terms. This doesn't seem to be possible, as the number of Goldbach's pairs increases with the magnitude of the sum (cf. Goldbach Comet), because of an underlying law. It is highly improbable that such law would cease to have effect from some particular number. Mathematical logic could even exclude it.