arXiv subjects

Berndt Gensel

Publications and source records attributed to Berndt Gensel.

5 recordsLinked to original sources

arXiv2023

Complex Circles of Partition and the Expansion Principles

In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to partitions defined within the complex plane, which now serves as both the base and bearing set. We employ the squeeze principle as a central tool for rigorously investigating the possibility to partition numbers with base set as a certain subset of the complex plane.

math.GM↗

arXiv2020

Studies in Additive Number Theory by Circles of Partition

In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that relates to deciding on the feasibility of partitioning numbers into certain subset of integers. In particular, our method allows us to partition any sufficiently large number $n\in\mathbb{N}$ into any set $\mathbb{H}$ with natural density strictly greater than $\frac{1}{2}$. This possibility could herald an unprecedented progress on categories of problems of similar flavour. The paper finishes by presenting an asymptotic proof of the binary Goldbach and Lemoine conjecture as an application of the developed method.

math.GM↗

arXiv2019

On The Infinitude of the Twin Primes

We present a novel approach to the Twin Prime Conjecture, basing on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the conjecture into the existence problem of such $x$. Using admissible residue classes modulo products of small primes and an adapted Selberg sieve, we partition the natural numbers into structured intervals $\mc{A}_n$, where the maximal possible prime divisor of $6x \pm 1$ is fixed. Within each $\mc{A}_n$, we apply the sieve to estimate the number of generator candidates that escape all local obstructions. Due to the \emph{parity problem} we cannot solve the problem with a Selberg sieve. It requires other sieves or methods. The author is searching for them and invites all interested people to help.

math.GM↗

arXiv2019

The Prime Number Formula of Gandhi

With the formula of Gandhi you can determine the on $p_n$ immedately subsequent prime $p_{n+1}$ from the knowledge of the primes $p_1, p_2, ... , p_n$. An elementary proof of its trueness will be detailed shown in this paper. Finally the question for the order of the primes on the number line will be discussed.

math.GM↗

arXiv2017

The Asymptotic Binary Goldbach and Lemoine Conjectures

In this paper, we use the former of the authors developed theory of \emph{circles of partition} to investigate possibilities to prove the binary Goldbach and Lemoine conjectures. We state the \emph{squeeze principle} and its consequences when the set of all odd prime numbers is the base set. Using this tool, we can prove asymptotic versions of the binary Goldbach and the Lemoine conjecture.

math.NT↗