Jamel Ghanouchi, A Proof of the Veracity of both Goldbach and De Polignac Conjectures, BMSA Volume 6, Bulletin of Mathematical Sciences and Applications (Volume 6)
https://www.scipress.com/BMSA.6.10
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 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 ³ 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.
Keywords:
    Algebraic Proof, De Polignac, Goldbach, Twin Primes