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A Proof of the Veracity of both Goldbach and De Polignac Conjectures

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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.

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Periodical:
Bulletin of Mathematical Sciences and Applications (Volume 6)
Pages:
10-20
Citation:
J. Ghanouchi, "A Proof of the Veracity of both Goldbach and De Polignac Conjectures", Bulletin of Mathematical Sciences and Applications, Vol. 6, pp. 10-20, 2013
Online since:
November 2013
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References:

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[2] D. R. Heath-Brown, J. C. Puchta, 2002, Integers represented as a sum of primes and powers of two. The Asian Journal of Mathematics, 6, no. 3, pages 535-565.

DOI: https://doi.org/10.4310/ajm.2002.v6.n3.a7

[3] H.L. Montgomery, Vaughan, R. C., 1975, The exceptional set in Goldbach's problem. Collection of articles in memory of Jurii Vladimirovich Linnik. Acta Arith. 27, 353–370.

[4] J. Richstein, 2001, Verifying the goldbach conjecture up to 4· 1014, Math. Comp., 70:236 ,1745--1749.

DOI: https://doi.org/10.1090/s0025-5718-00-01290-4

[5] L. E. Dickson, 2005, History of The Theory of Numbers, Vol1, New York Dover.

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