Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions
Ameneh Farhadian
TL;DR
The paper reframes Goldbach in a distributional, probabilistic setting, showing that for large even numbers the chance of a counterexample is extraordinarily small if Goldbach holds up to $2N$. By constructing $A_n$ and $B_n$ from prime distributions, it derives a bound on the failure probability $\\P(n)=\\exp(-n/\\ln^2 n)$ and aggregates to $\\P<e^{-N^\\a}$ with $\\a=1-\frac{2\\ln\\ln N}{\\ln N}$. With numerical verification of Goldbach up to $4\\times10^{18}$, this yields an astronomically small upper bound $<e^{-10^{15}}$, effectively ruling out counterexamples in practice. The authors further generalize the conjecture to prime-like subsets obtained via random $\\pm1$ shifts of primes and verify the generalized conjecture up to $2\\times10^8$ even numbers, demonstrating the robustness and applicability of a distribution-focused perspective.
Abstract
Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails. Assuming the conjecture holds true for all even number less than $2N$, we prove this probability is less than $e^{-N^α}$, where $ α= 1 - \frac{2\ln\ln N}{\ln N}$. For large $N$, this probability becomes vanishingly small, effectively precluding the existence of counterexamples in practice. If $N =4 \times 10^{18}$, the probability of a counterexample is less than $e^{-10^{15}}$. Our approach fundamentally depends on the distributional properties of primes rather than their primality per se. This perspective enables a natural generalization of the conjecture to non-prime subsets of integers that exhibit similar distributional characteristics. As a concrete example, we construct new subsets by applying random $\pm 1$ shifts to primes, which preserve the essential prime-like distributional properties. Computational verification confirms that this generalized Goldbach conjecture holds for all even integers up to $2 \times 10^{8}$ within these modified subsets.