## Experimental Mathematics

### Jumping champions

#### Abstract

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 $10^{12}$, 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 $x=1$.$7427\cdot10^{35}$, 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 consecutive primes are formidable enough that even that strong conjecture does not suffice to produce a rigorous proof about the behavior of jumping champions.

#### Article information

Source
Experiment. Math., Volume 8, Issue 2 (1999), 107-118.

Dates
First available in Project Euclid: 12 March 2003

https://projecteuclid.org/euclid.em/1047477055

Mathematical Reviews number (MathSciNet)
MR1700573

Zentralblatt MATH identifier
0993.11045

Subjects
Primary: 11Y70: Values of arithmetic functions; tables
Secondary: 11N05: Distribution of primes

#### Citation

Odlyzko, Andrew; Rubinstein, Michael; Wolf, Marek. Jumping champions. Experiment. Math. 8 (1999), no. 2, 107--118. https://projecteuclid.org/euclid.em/1047477055