Divisor Goldbach Conjecture and its Partition Number

TL;DR

This paper extends the Goldbach conjecture to a broader class of integers, proposing that any positive integer with a divisor can be expressed as a sum of specific prime numbers related to its divisors, supported by preliminary computational verification.

Contribution

It introduces a generalized form of the Goldbach conjecture based on divisibility, providing initial proofs and computational evidence for its validity.

Findings

No counterexamples found in preliminary computational checks.

The conjecture holds for various tested cases.

Provides initial proofs for special cases.

Abstract

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on the divisible factor of this integer, that is: For any positive integer $n~(n>2)$, if there exists an integer $m$, such that $m|n~( 1 < m < n )$, then $n=\sum_{i=1}^m p_i $, where $ p_i~(i=1,2 ,3...m)$ is prime number. In addition, for more prime summands, the combinatorial counting is also discussed. For some special cases, some brief proofs are given. By the use of computer, the preliminary numerical verification was given, there is no an anti-example to be found.