A PROOF OF THE VERACITY OF BOTH GOLDBACH AND DE POLIGNAC CONJECTURES

Abstract The present algebraic development begins by an exposition of the data of the problem. The definition of the primal radius r 0 is : For all positive integer x 3 exists a finite number of integers called the primal radius r 0 , for which x r and x r are prime numbers. The corollary is that 2x (x r) (x r) is always the sum of a finite number of primes. Also, for all positive integer x 0 , exists an infinity of integers r 0 , for which x r and r x are prime numbers. The conclusion is that 2x (x r) (r x) is always an infinity of differences of primes.


Introduction
There is a similarity between the assertion : "an even number is always the sum of two primes" and the assertion : "an even number is always the difference of two primes". The present article gives the proof that the two assertions are the consequences of the same concept by the introduction of the notion of the primal radius.

The proof
Let us suppose that exists an integer for which 2x is never the sum of two primes, then for all

Lemma 1
The following formula Imply that exist p 1 and p 2 prime numbers for which b = 0

Proof of lemma 1
If x is prime 2x = x + x is the sum of two primes, then p1  p2  0 We will suppose firstly that Let

Thus
We pose with a different of zero Let It means b=0 thus are primes y = r is the primal radius. As there is the condition p 2  x  p 1 , there is not an infinity of p 2  p 1 With the same reasoning and calculus b 0 But b can not be equal to zero in all cases, it means there is an impossibility related to the fact that the conjecture is indecidable and if it is so, it is true ! Because, we would find in the case it is indecidable and false with the computer the 2x different of all sum of primes and it is contradictory !
Now, if we suppose that for all p 2  p 1 primes, exixts with the same reasoning, the same calculus but replacing x by y and y by x , we prove that b  0 , which means that for all positive integer x , exists  p 1, p 2 for which y is the primal radius. As there is no condition on x , y , p 1, p 2 there is an infinity of couples of primes ( p 1, p 2 ) For x = 1 , p 1, and p 2 are twin primes. Let us prove it. Let us suppose that exists an integer x  0 for which 2x is never the difference of two primes, then for all p 1 and p 2 primes, But for all p 1, p 2 exists y , for which Let We deduce that

Lemma 2
The following formula Imply that exist p 1 and p 2 prime numbers for which b 0

Proof of lemma 2
If x is prime 0xx is the sum of two primes, then We will suppose firstly that

Bulletin of Mathematical Sciences and Applications Vol. 6 17
Another proof : let u,u',v,v' verifying

Thus
We pose with a different of zero

Volume 6
Let up 1 vp 2 hence Let 2u hence It means b=0 thus are primes y = r is the primal radius. As there is no condition, there is an infinity of p 1, p 2.
With the same calculus and reasoning, it implies that b = 0 . But b can not be equal to zero in all cases, it means there is an impossibility related to the fact that the conjecture is indecidable and if it is so, it is true ! Because, we would find in the case it is indecidable and false with the computer the 2x different of all sum of primes and it is contradictory ! For is a difference of an infinity of couples of primes. There is an infinity of consecutive primes. And for all exists primes for which

Conclusion
The notion of the primal radius as defined in this study allows to confirm that for all integer x  3 exists a number r  0 for which x  r and x  r are primes and that for all integer x  0 exists a number r  0 for which x  r and r  x are primes and that exists an infinty of such primes. r is called the primal radius The corrolary is the proof of the Goldbach conjecture and de Polignac conjecture which stipulate, the first that an even number is always the sum of two prime numbers, the second that an even number is always the difference between two primes and that there is an infinity of such couples of primes. Another corollary is the proof of the twin primes conjecture which stipulates that there is an infinity of consecutive primes.