Title: Arbitrarily Large Goldbach (Even) Integers.
Abstract: On investigating the subject of partitions on a prescribed subset of integers, Chen proved in 1966 that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. This result further gave a giant leap towards the proof of Goldbach Conjecture, even as the conjecture still remained unproved. To make some improvements on this, we do not exactly prove the Goldbach Conjecture. But using Dirichlet's Theorem, Green-Tao Theorem, Equivalences on prime-pairs, etc., we associate Green-Tao and Dirichlet's Theorems with the even integers m and the partition function w(m) of m, in a more generalized way. We prove that for any A.P., A containing primes (as defined by Green-Tao and Dirichlet's Theorems respectively), there exist a corresponding even integer m such that every prime in the solution set (and in the partition function) of m are terms of A; and its converse. We prove that lim sup w(m) is unbounded; that is, that there exist even integers that can be written in an arbitrarily large number of ways as a sum of two primes. And hence that there exist arbitrarily large Goldbach even integers. Goldbach here being an adjective for its primes partitions.
Publication Year: 2021
Publication Date: 2021-11-22
Language: en
Type: preprint
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