William, it only solves world hunger sofar... :-D

My reasoning in words is as follows:

Assuming that Levy's conjecture, that any odd number equal or larger than 7 can be produced by the sum of "twice a prime (P) plus a prime", is indeed correct, I try to assess what the implications could be.

Here's just one. Since all all odd numbers (O) > 7 also comprise all the prime numbers, then also all odd prime numbers (except 2,3 and 5) could be produced by the sum of "twice a prime plus a prime".

Since Goldbach conjectured that all even numbers (E) larger than or equal to 4 are the sum of two primes, we can also write this equation as (twice a prime + a prime) + (twice a prime + a prime). This gives you the Goldbach conjecture decomposed into 4 prime numbers (of which some could be the same of course).

My first assumption is that if you assume you can make all odd numbers from "twice a prime plus a prime", then it should be easy to prove that the sum of two these odd numbers can produce any even number (i.e. the conclusion is that with 4 primes you can produce all even numbers).

But that's still four primes and I would like to get down to two. I then tried to find a way to express the last two primes (indexed by k,l) in the "4-prime Goldbach equation" into the first two primes (indexed by i,j).

One connection I tried is that there must be i,j's and k,l's that produce odd numbers that are two apart. It should be true for all odd numbers produced by "primes i,j + 2". that there is an equal odd number produced by primes k,l. I therefore can express the i,j's in the k,l's and don't need the k,l's (although some doubt about the correctness of this step is slumbering in my head...).

Example:

2*13 + 7 = 33

2*2 + 31 = 33+2

2*13+ 7 + 2 = 2*2 + 31

Other ways I thought off with potential to reduce the 4 primes to less?

1. Maybe a trick with

and

. If we combine these in the Goldbach conjecture then the sum becomes

. Doesn't produce all even numbers of course, but maybe we could relate it back to the k,l primes.

2. Maybe there's a trick with

and

with the Goldbach sum becoming

. Again not sure if leads anywhere.

Boarding on a trip to Rio now, so some more time to think about all of this. I don't preclude that the conclusion could be that my reasoning contains some major flaws...

But hey, progress can only be made through a process of creative destruction. 8)

P.S.

Although my biggest nightmare is of course to find a rock solid prove of the Goldbach conjecture at 30000 feet above the ocean, but then without any internet connection and suddenly the pilot announcing some serious engine problems... I vaguely recall a similar story about a postcard, a stormy boat trip and an alleged proof of the Riemann hypothesis?