The jumping champion conjecture

An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. For the $n$th prime $p_{n}$, the $n$th primorial $p_{n}^{\sharp}$ is defined as the product of the first $n$ primes. In 1999, Odlyzko, Rubinstein and Wolf provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $p_{1}^{\sharp}, p_{2}^{\sharp}, p_{3}^{\sharp}, p_{4}^{\sharp}, p_{5}^{\sharp}, ...$, that is, $2, 6, 30, 210, 2310, ....$ In this paper, we prove that an appropriate form of the Hardy-Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$.Comment: 19 pages, 1 tabl

On the equidistribution properties of patterns in prime numbers Jumping Champions, metaanalysis of properties as Low-Discrepancy Sequences, and some conjectures based on Ramanujan's master theorem and the zeros of Riemann's zeta function

The Paul Erd\H{o}s-Tur\'an inequality is used as a quantitative form of Weyl' s criterion, together with other criteria to asses equidistribution properties on some patterns of sequences that arise from indexation of prime numbers, Jumping Champions (called here and in previous work, "meta-distances" or even md, for short). A statistical meta-analysis is also made of previous research concerning meta-distances to review the conclusion that meta-distances can be called Low-discrepancy sequences (LDS), and thus exhibiting another numerical evidence that md's are an equidistributed sequence. Ramanujan's master theorem is used to conjecture that the types of integrands where md's can be used more succesfully for quadratures are product-related, as opposite to addition-related. Finally, it is conjectured that the equidistribution of md's may be connected to the know equidistribution of zeros of Riemann's zeta function, and yet still have enough "information" for quasi-random integration ("right" amount of entropy).Comment: 13 pages, 7 Figures, 4 table