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On the binary Goldbach conjecture under primorial-based constraints

2024; RELX Group (Netherlands); Linguagem: Inglês

10.2139/ssrn.4839110

1556-5068

R Laniewski,

Finite Group Theory Research

This paper introduces a novel, mathematically justified constraint on the binary Goldbach conjecture by scaling even integers with successive odd primes, thus excluding these scaling factors as potential summands (p, q). By introducing a lower bound on the prime number candidates and adapting the circle method to accommodate variable lower and upper bounds, this approach dramatically alters the statistical landscape of the problem. We computationally verify the binary Goldbach conjecture for even integers of the form c = 2 x p_n# (an even number multiplied by a primorial) up to n = 5134, where c is on the order of 10^21,603 (21,604 digits). Beyond n > 5134, extensive computations fail to find prime pairs that sum to c. Verifying or refuting these potential counterexamples presents a significant challenge due to their large sizes (on the order of 10^34,605 and higher). This paper probes both the theoretical framework and the computational methods for testing the limits of the binary Goldbach conjecture.