On Jun 14, 5:28 am, mluttgens <lutt...@gmail.com> wrote: > In P, the number 26 = 3+23, 5+21, 7+19, 9+17, 11+15, 13+13, 15+11, > 17+9, 19+7, 21+5 and 23+3, all the terms coming from the successive > uneven numbers of S and S' from 3 to 23, where 23 = 26 - 3. > As S and S' contain all the prime numbers inferior or equal to 26-3 , > the even number 26 is necessarily the sum of two primes. > Let?s note that it is not necessary to indentify those primes. Knowing > that they exist in S and S? is enough..
The argument you give here is fallacious. The sets S and S' also contain odd numbers that are not prime. This makes it necessary to identify the particular primes and it is not enough to know that primes exist in S and S'. In your example for 26, it is not possible to use the sums 5+21, 9+17, 11+15, 15+11, 17+9, 21+5 since each contains a composite integer. My earlier question to you was: what if there is an integer for which _all_ the sums are inadmissable, not just some as in the case of 26? Nothing in your argument rules out this possibility.
If a counterexample exists it will have to be larger that 10^17 because computer verifications have been performed up to about that value. A correct proof is highly likely to require some highly technical innovations.