On differences between consecutive numbers coprime to primorials

2020/07/03 by Mario Ziller, Ziller, Mario
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Mathematics and Applications #math.NT

paper · pdf · doi:10.48550/arxiv.2007.01808

14 pages, 1 table

arxiv created 2020/07/03 · arxiv updated 2020/07/06

Abstract

We consider the ordered sequence of coprimes to a given primorial number and investigate differences between consecutive elements. The Jacobsthal function applied to the concerning primorial turns out to represent the greatest of these differences. We will explore the smallest even number which does not occur as such a difference. Little is known about even natural numbers below the respective Jacobsthal function which cannot be represented as a difference between consecutive numbers coprime to a primorial. Existence and frequency of these numbers have not yet been clarified. Using the relation between restricted coverings of sequences of consecutive integers and the occuring differences, we derive a bound below which all even natural numbers are differences between consecutive numbers coprime to a given primorial p_k#. Furthermore, we provide exhaustive computational results on non-existent differences for primes p_k up to k=44. The data suggest the assumption that all even natural numbers up to h(k-1) occur as differences of coprimes to p_k# where h(n) is the Jacobsthal function applied to p_n#.

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