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 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. Erdős 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.2010 Mathematics Subject Classification. Primary 11N05. Secondary 11P32, 11N36.

The Jumping Champion Conjecture

“…Very recently, we have extended their method to give a complete proof of Conjecture 2, again under the same assumption. (See Goldston and Ledoan [5]. )…”

“…Applying (1) with k = 3 and the estimate S({0, d ′ , d}) = O((log d) 2 ) (see (3.3) as proved in Section 4 of Goldston and Ledoan [5], or see ( 16) below for a sharper estimate), we obtain…”

“…We shall now prove (3) when d is confined to the range 2 ≤ d ≤ log x/(log log x) 5 . By ( 5) and ( 6), we have…”

“…for even integers d, which leads us to the inequality S(d) < S(p ♯ k ), valid for every integer d in the range 2 ≤ d < p ♯ k . (See Lemma 2.2 in Goldston and Ledoan [5].) The primorials have a very simple distribution governed by the prime number theorem which, from the alternative formulation…”

“…Our theorem is a simple deduction from Lemmas 3, 4 and 5 below. Before establishing these results, we retain the notation of the proof of Theorem 2.1 in Goldston and Ledoan [5] for the floor function with respect to a given increasing sequence {a j } ∞ j=1 and introduce a similar notation for the ceiling function. We write ⌊y⌋ aj = a n if a n ≤ y < a n+1 , and ⌈y⌉ aj = a n if a n−1 < y ≤ a n .…”