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JUMPING CHAMPIONS AND GAPS BETWEEN CONSECUTIVE PRIMES

2011; World Scientific; Volume: 07; Issue: 06 Linguagem: Inglês

10.1142/s179304211100471x

1793-0421

D. A. Goldston, Andrew Ledoan,

graph theory and CDMA systems

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.