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Jumping champions. (English) Zbl 0993.11045
An integer \(D\) is a “jumping champion” if there is some \(x\geq 2\) such that \(D\) is a most frequent value of \(p_{n+1}- p_n\) for primes \(p_n\leq x\). For values of \(x\leq 1000\) the jumping champion is sometimes 2, sometimes 4 and sometimes 6; and sometimes two or more of these are equally frequent. However between \(10^3\) and \(10^{12}\) the jumping champion is always 6. The paper presents heuristic and numerical evidence suggesting that the jumping champion is always 6 from 1000 to about \(1.7\times 10^{35}\), but is then replaced by 30. It appears that 30 is eventually replaced by 210, and then by 2310, and indeed that the jumping champions are precisely 4 and the “primorial” numbers \(\prod_{j\leq k}p_j\).
The heuristics are based on a quantitative form of the Hardy-Littlewood prime \(k\)-tuples conjecture. There are a number of interesting plots of numerical data, corroborating the heuristics.

MSC:
11N05 Distribution of primes
11Y70 Values of arithmetic functions; tables
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