# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bombieri E., Proc. Roy. Soc. Ser. A 293 pp 1– (1966) · Zbl 0151.04201 [2] Brent R. P., Math. Comp. 28 pp 315– (1974) [3] DOI: 10.1090/S0025-5718-1975-0369287-1 [4] Erdos P., Elem. Math. 35 (5) pp 115– (1980) [5] Gallagher P. X., Mathematika 23 (1) pp 4– (1976) · Zbl 0346.10024 [6] Guy R. K., Unsolved problems in number theory, (1994) · Zbl 0805.11001 [7] Halberstam H., Sieve methods (1974) · Zbl 0298.10026 [8] DOI: 10.1007/BF02403921 · JFM 48.0143.04 [9] Harley R., ”Some estimates by Richard Brent applied to the” high jumpers ”problem” (1994) [10] Ingham A. E., The distribution of prime numbers (1932) · Zbl 0006.39701 [11] DOI: 10.1007/BFb0060851 [12] Nelson H., J. Rec. Math. 11 pp 231– (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.