A Simple Explanation for the Goldbach Conjecture

TL;DR

This paper proposes a probabilistic explanation for the Goldbach Conjecture, suggesting that its validity extends to subsets of natural numbers with similar distributions to primes, supported by computational verification up to 200 million.

Contribution

It introduces a probabilistic perspective on the Goldbach Conjecture, extending its applicability to various number subsets with similar distributions to primes, supported by computational evidence.

Findings

Goldbach conjecture holds for reconstructed sets similar to primes

Probability, not number theory, may explain the conjecture's validity

Computational verification up to 2×10^8 supports the generalization

Abstract

In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is similar to the prime numbers is negligible. This result makes it possible to generalize the conjecture to any subset of natural numbers whose distribution is similar to the prime numbers. Additionally, we selected several new subsets whose distribution amongst the natural numbers are similar to the prime numbers by randomly addition of +1 and -1 to the prime numbers and checked the Goldbach conjecture for every even integer less than $2 \times 10^8$ by computer. As it was expected, the Goldbach conjecture holds true for these new reconstructed sets, as well. Consequently, the conjecture can be generalized to any subset of natural numbers whose distribution is similar to the prime numbers. That is "being prime" is not necessary for the conjecture to hold for the instances. This fact brings to mind the idea that perhaps what makes the Conjecture to hold for the instances is "probability", not number theory facts.