An Extension Of Vinogradov's Theorem

TL;DR

This paper extends Vinogradov's Theorem to include all sufficiently large natural numbers, both even and odd, as sums of three primes, and explores its implications for the Goldbach Conjecture.

Contribution

It generalizes Vinogradov's Theorem to cover all large natural numbers and establishes connections with the Goldbach Conjecture and prime partition functions.

Findings

Vinogradov's Theorem is extended to all large natural numbers.

The paper derives bounds for the size of numbers for which the theorem holds.

It demonstrates the equivalence and implications between Vinogradov's Theorem and the Goldbach Conjecture.

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).