Sifting Function Partition for the Goldbach Problem

TL;DR

This paper introduces new sieve methods that reduce the need to sift out all composite numbers in the Goldbach problem, focusing instead on leaving prime numbers in the residuals to prove the conjecture.

Contribution

It proposes novel sifting function partition techniques that simplify the sieve process and potentially solve the Goldbach and twin prime problems.

Findings

New sieve methods reduce the complexity of sifting in Goldbach problem

Residual primes can be used to establish Goldbach conjecture

Methods eliminate indeterminacy of sifting functions

Abstract

All sieve methods for the Goldbach problem sift out all the composite numbers; even though, strictly speaking, it is not necessary to do so and which is, in general, very difficult. Some new methods introduced in this paper show that the Goldbach problem can be solved under sifting out only some composite numbers. In fact, in order to prove the Goldbach conjecture, it is only necessary to show that there are prime numbers left in the residual integers after the initial sifting! This idea can be implemented by using one of the three methods called sifting function partition by integer sort, sifting function partition by intervals and comparative sieve method, respectively. These are feasible methods for solving both the Goldbach problem and the problem of twin primes. An added bonus of the above methods is the elimination of the indeterminacy of the sifting functions brought about by their upper and lower bounds.