Jumping champions and gaps between consecutive primes

TL;DR

This paper investigates the nature of jumping champions among primes, providing a proof that a conjecture about their divisibility properties follows from a well-known prime pair conjecture, advancing understanding of prime gaps.

Contribution

It establishes that the divisibility conjecture for jumping champions is a consequence of the Hardy-Littlewood prime pair conjecture.

Findings

Proves the divisibility conjecture assuming the prime pair conjecture.

Extends a method from Erdős and Straus (1980) to analyze prime gaps.

Links the behavior of jumping champions to deep prime distribution conjectures.

Abstract

The most common difference that occurs among the consecutive primes less than or equal to $x$ is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given $x$. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf provided heuristic and empirical evidence in support of the conjecture that the numbers greater than 1 that are jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,... As a step towards proving this conjecture they introduced a second weaker conjecture that any fixed prime $p$ divides all sufficiently large jumping champions. In this paper we extend a method of P. Erd\H{o}s and E. G. Straus from 1980 to prove that the second conjecture follows directly from the prime pair conjecture of G. H. Hardy and J. E. Littlewood.