About the logic of the prime number distribution

TL;DR

This paper explores the distribution of prime numbers through two key sequences, proposing a wave interference model linked to the Riemann Zeta Function and highlighting the significance of the number 5.

Contribution

It introduces a novel wave interference model based on two prime-related sequences to explain prime distribution and its connection to the Riemann Zeta Function.

Findings

Prime numbers are mainly in two sequences: 6n+1 and 6n+5.

Non-prime numbers result from sequence recurrences similar to harmonic overtones.

The distribution can be visualized with a wave interference model.

Abstract

There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the form 6n+1. All existing prime numbers seem to be contained in these two number sequences, except of the prime numbers 2 and 3. Riemanns Zeta Function also seems to indicate, that there is a logical connection between the mentioned number sequences and the distribution of prime numbers. This connection is indicated by lines in the diagram of the Zeta Function, which are formed by the points s where the Zeta Function is real. Another key role in the distribution of the prime numbers plays the number 5 and its periodic occurrence in the two number sequences SQ1 and SQ2. All non-prime numbers in SQ1 and SQ2 are caused by recurrences of these two number sequences with increasing wave-lengths in themselves, in a similar fashion as Overtones (harmonics) or Undertones derive from a fundamental frequency. On the contrary prime numbers represent spots in these two basic Number Sequences SQ1 and SQ2 where there is no interference caused by these recurring number sequences. The distribution of the non-prime numbers and prime numbers can be described in a graphical way with a -Wave Model- (or Interference Model) -- see Table 2.