Jumping champions

The asymptotic frequency with which pairs of primes below x differ by some fixed integer is understood heuristically, although not rigorously, through the Hardy-Littlewood k-tuple conjecture. Less is known about the differenc es of consecutive primes. For all x between 1000 and 10(12), the most commo n difference between consecutive primes is 6. We present heuristic and empi rical evidence that 6 continues as the most common difference (jumping cham pion) up to about x = 1.7427 . 10(35), where it is replaced by 30. In turn, 30 is eventually displaced by 210, which is then displaced by 2310, and so on. Our heuristic arguments are based on a quantitative form of the Hardy- Littlewood conjecture. The technical difficulties in dealing with consecuti ve primes are formidable enough that even that strong conjecture does not s uffice to produce a rigorous proof about the behavior of jumping champions.