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/17392149. 1 Introduction The Goldbach conjecture, one of the oldest and most famous unsolved problems in number theory, states that every even integer greater than 2 can be expressed as the sum of two primes. While this conjecture remains unproven despite extensive computational verification and partial results, it has profound implications for our understanding of prime distribution. In this note, we formalize the observation that the Goldbach conjecture implies a regularity condition on the distribution of primes in linear intervals, which we call the “Prime-inEvery-Slice” conjecture. We then demonstrate that this intermediate condition immediately implies two other classical conjectures: Legendre’s conjecture concerning primes between consecutive perfect squares, and Oppermann’s conjecture concerning primes in specific subintervals. 2 The Prime-in-Every-Slice Conjecture Definition 2.1 (Prime-in-Every-Slice).For every integer ν≥2, consider the intervals ((k− 1)ν, kν] for k= 1,2, . . . , ν + 1. We conjecture that each such interval contains at least one prime number. 1 This conjecture essentially states that primes are distributed with sufficient regularity that no gap larger than νexists among primes up to (ν+ 1)ν. It is stronger than Bertrand’s postulate, which guarantees a prime in (n, 2n] for n≥1. Proposition 2.2. If the Goldbach conjecture holds, then the Prime-in-Every-Slice conjecture is true. Proof. Suppose for contradiction that there exists νand ksuch that the interval ((k−1)ν, kν] contains no prime. Consider the even integer N= (k−1)ν+kν = (2k−1)ν. By Goldbach’s conjecture, Nmust be expressible as the sum of two primes: N=p+q for some primes pand q. Without loss of generality, assume p≤q. If p≤(k−1)ν, then q=N−p≥(2k−1)ν−(k−1)ν=kν. But also q=N−p<(2k−1)ν, so (k−1)ν < q ≤kν, meaning q∈((k−1)ν, kν]. However, by assumption, this interval contains no primes, yielding a contradiction. Therefore, if Goldbach holds, no such gap can exist, and each interval must contain at least one prime. 3 Consequences for Classical Conjectures 3.1 Legendre’s Conjecture Conjecture 3.1 (Legendre, 1808).For every positive integer n, there exists at least one prime in the interval [n2,(n+ 1)2]. Theorem 3.2. The Prime-in-Every-Slice conjecture implies Legendre’s conjecture. Proof. Let ν=n. The interval [n2,(n+ 1)2] has length (n+ 1)2−n2= 2n+ 1, which can be covered by at most three intervals of length n: [n2, n2+n],[n2+n, n2+ 2n],[n2+ 2n, (n+ 1)2]. By the Prime-in-Every-Slice conjecture with ν=n, each interval of length ncontains at least one prime. Therefore, the interval [n2,(n+ 1)2] contains at least one prime. 3.2 Oppermann’s Conjecture Conjecture 3.3 (Oppermann, 1882).For every positive integer n > 1, there exists at least one prime in the interval [n2, n(n+1)] and at least one prime in the interval [n(n+1),(n+1)2]. Theorem 3.4. The Prime-in-Every-Slice conjecture implies Oppermann’s conjecture. Proof. Observe that: •The interval [n2, n(n+ 1)] has length n(n+ 1) −n2=n •The interval [n(n+ 1),(n+ 1)2] has length (n+ 1)2−n(n+ 1) = n+ 1 By choosing ν=n+1, the Prime-in-Every-Slice conjecture guarantees that each interval of length νcontains at least one prime. Since both intervals have length at most ν, each must contain at least one prime. 2 4 Conclusion We have established the following logical chain of implications: Goldbach conjecture ⇒Prime-in-Every-Slice conjecture ⇒Legendre and Oppermann conjectures This observation provides a unified perspective linking these classical conjectures and demonstrates how the truth of the Goldbach conjecture would enforce a strong form of regularity in the distribution of prime numbers. Specifically, if Goldbach is true, it necessitates that primes appear with sufficient frequency in linear intervals to automatically satisfy the conditions required by both Legendre’s and Oppermann’s conjectures concerning primes in quadratic intervals. The significance of this result lies not in proving these conjectures, but in revealing their structural interconnection. The Prime-in-Every-Slice conjecture serves as a bridge, showing that additive properties of primes (as expressed in Goldbach) have direct implications for their multiplicative distribution patterns (as expressed in Legendre and Oppermann). This logical architecture suggests that progress on any of these problems may yield insights applicable to the others. 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] P. Ribenboim, The Little Book of Bigger Primes, Springer, 2004. [4] M. Sallis, A Prime in Every Slice: Beyond Bertrand’s Postulate, Zenodo, 2025. https: //zenodo.org/records/17392149 3