Consequences of the Goldbach Conjecture for Classical Prime Distribution Conjectures

Consequences of the Goldbach Conjecture for Classical Prime Distribution Conjectures Morfis Sallis University of Ioannina [email protected] November 2, 2025 Abstract We observe that under the assumption that the Goldbach conjecture holds, certain classical conjectures on the distribution of prime numbers follow as immediate consequences. Specifically, we demonstrate that Goldbach ⇒”Prime in Every Slice” (our proposed conjecture) ⇒Legendre’s conjecture and Oppermann’s conjecture. The logical chain provides a unified perspective connecting these independent conjectures. The original formulation of the Prime-in-Every-Slice conjecture is available at Zenodo: [https://zenodo.org/records/17507829](https://zenodo.org/records/17507829). 1 Introduction The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. While unproven, it has profound implications for the distribution of primes. In this note, we formalize the observation that the Goldbach conjecture implies a regularity condition on the distribution of primes in linear intervals, which in turn implies two classical conjectures, namely Legendre’s and Oppermann’s conjectures. 2 The Prime-in-Every-Slice Conjecture [Prime-in-Every-Slice] For every integer ν≥2, consider the intervals ((k−1)ν, kν] for k= 1,2, . . . , ν + 1. Then each such interval contains at least one prime. If the Goldbach conjecture holds, then the Prime-in-Every-Slice conjecture is true. Proof. Suppose, for contradiction, that there exist integers ν≥2 and k∈1,2,...,ν+ 1 such that the interval ((k−1)ν, kν] contains no prime. Define N= 2(k−1)ν+ 1. 1 Then Nis an even integer greater than 2. By the Goldbach conjecture, there exist primes p and qsuch that N=p+q. Note that (k−1)ν < (k−1)ν+ 1 ≤kν, so (k−1)ν+ 1 ∈((k−1)ν, kν]. Assume the interval ((k−1)ν, kν] contains no prime. Then every prime is either ≤(k−1)ν or ≥kν + 1. If p≤(k−1)νand q≤(k−1)ν, then p+q≤2(k−1)ν < 2((k−1)ν+ 1) = N, a contradiction. If p≥kν+1 and q≥kν+1, then p+q≥2(kν+1) >2kν ≥2((k−1)ν+1) = N, also a contradiction. Thus, one of por qmust lie in ((k−1)ν, kν], contradicting the assumption that this interval contains no prime. 3 Consequences for Legendre and Oppermann 3.1 Legendre’s Conjecture [Legendre] For every positive integer n, there exists at least one prime in [n2,(n+ 1)2]. The Prime-in-Every-Slice conjecture implies Legendre’s conjecture. Proof. Let ν=n. Divide [n2,(n+ 1)2] into two intervals of length n: [n2, n2+n] and [n2+n, (n+1)2]. By the Prime-in-Every-Slice conjecture, each interval contains at least one prime. Therefore, the interval [n2,(n+ 1)2] contains at least one prime. 3.2 Oppermann’s Conjecture [Oppermann] For every positive integer n, there exists at least one prime in [n2, n(n+ 1)] and at least one prime in [n(n+ 1),(n+ 1)2]. The Prime-in-Every-Slice conjecture implies Oppermann’s conjecture. Proof. The two intervals in Oppermann’s conjecture have length neach. By choosing ν= n, the Prime-in-Every-Slice conjecture guarantees that each interval contains at least one prime. 4 Conclusion Under the assumption that the Goldbach conjecture holds, we have established that the Prime-in-Every-Slice conjecture is true, which in turn implies the validity of both Legendre’s and Oppermann’s conjectures. This logical chain demonstrates that the truth of Goldbach enforces a regular distribution of prime numbers in linear intervals, automatically satisfying classical conjectures concerning primes in quadratic intervals. This provides a unified perspective connecting these long-standing problems in number theory. 2 References 1. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, Oxford University Press, 2008. 2. T. Nagell, Introduction to Number Theory, 2nd edition, Wiley, 1951. 3. Ribenboim, The Little Book of Bigger Primes, Springer, 2004. 4. Morfis Sallis, Prime-in-Every-Slice Conjecture, Zenodo, 2025. [https://zenodo.org/ records/17507829](https://zenodo.org/records/17507829) 3