Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912337458364416 |
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| author | Farhadian, Ameneh |
| author_facet | Farhadian, Ameneh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions Farhadian, Ameneh Number Theory Combinatorics 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. |
| title | Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2504.14353 |