JUMPING CHAMPIONS AND GAPS BETWEEN CONSECUTIVE PRIMES

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 Odlyzko, Rubinstein and 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 toward 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 Erdős and Straus from 1980 to prove that the second conjecture follows directly from the prime pair conjecture of Hardy and Littlewood.