An Extension Of Vinogradov's Theorem
Abstract
n 1937 Ivan Vinogradov proved the three prime sum version of the Goldbach Conjecture, often called the weak form of Goldbach Conjecture. And that it holds for "sufficiently large" odd natural numbers. In this work we use Dirichlet Theorem, Modulo Arithmetics, etc. to extended Vinogradov's Theorem such that every sufficiently large natural number (both even and odd) can be expressed as a sum of three primes. We highlight the configuration of primes for any special case of the three prime sum. Hence we obtain Vinogradov's Theorem as a special case of this extended version. We show how Vinogradov's Theorem implies the Goldbach Conjecture; how it (Vinogradov's Theorem) can be derived from it and vice versa. We also obtain the lower bound of the sufficiently largeness. And concluding, we highlight some relationships between the partition function of the Vinogradov's integer w(v) and the Goldbach integer w(m).
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07334
 Bibcode:
 2021arXiv211007334U
 Keywords:

 Mathematics  General Mathematics;
 11P32
 EPrint:
 10 Pages,v5 extends Vinogradov theorem, v4 proves Goldbach Conjecture, v3 proves the existence arbtrarily large m that is Goldbach, it also generalizes v2. v2 focuses on relationships between the partition function w(m) of m and the distribution of primes in PAP of length 3, and the number of distinct such P.A.Ps. as a function of w(m). v1 is the foundation