arXiv subjects

John F. Morack

Publications and source records attributed to John F. Morack.

3 recordsLinked to original sources

arXiv2017

A short note on the computation of the generalised Jacobsthal function for paired progressions

Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. We extended and adapted algorithms described for the computation of the common Jacobsthal function, and computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73. All these values fulfil the conjectured specific bound. In addition to this note, we provide a detailed review of the algorithmic approaches and the complete computational results in ancillary files.

math.NT↗

arXiv2017

Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs

We investigate progressions in the set of pairs of integers $\mathbb{Z}^2$ and define a generalisation of the Jacobsthal function. For this function, we conjecture a specific upper bound and prove that this bound would be a sufficient condition for the truth of the Goldbach conjecture, the infinitude of prime twins, and more general of prime pairs with a fixed even difference.

math.NT↗

arXiv2017

Algorithmic concepts for the computation of Jacobsthal's function

The Jacobsthal function has aroused interest in various contexts in the past decades. We review several algorithmic ideas for the computation of Jacobsthal's function for primorial numbers and discuss their practicability regarding computational effort. The respective function values were computed for primes up to 251. In addition to the results including previously unknown data, we provide exhaustive lists of all sequences of the appropriate maximum lengths in ancillary files.

math.NT↗