2025/04/19 by Farhadian, Ameneh
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2504.14353
Due to the distribution of primes among integers, we establish an upper bound for the probability ℙn 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-Nα, where α= 1 - (2lnln N)/(ln N). For large N, this probability becomes vanishingly small, effectively precluding the existence of counterexamples in practice. If N =4 × 1018, the probability of a counterexample is less than e^-1015. 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 ± 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 × 108 within these modified subsets.