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- A. Odlyzko, M. Rubinstein and M. Wolf, "Jumping champions," Experimental Math., 8:2 (1999) 107-118. MR 2000f:11164
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 differences of consecutive primes. For all x between 1000 and 1012, the most common difference between consecutive primes is 6. We present heuristic and empirical evidence that 6 continues as the most common difference (jumping champion) up to about x=1.7427· 1035, where it is replaced by 30. In turn, 30 is eventually displaced by 210, which then is 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 consecutive primes are formidable enough that even that strong conjecture does not suffice to produce a rigorous proof about the behavior of jumping champions.