Family of prime-representing constants: use of the ceiling function

TL;DR

This paper introduces a new family of prime-representing constants using the ceiling function, providing a recursive relation that generates all known primes and demonstrating their irrationality.

Contribution

It proposes a novel recursive relation involving the ceiling function to generate all primes and introduces a new family of irrational prime-representing constants.

Findings

The recursive relation successfully generates all known primes.

The constants h_n are proven to be irrational.

The approach extends previous prime-generating methods using floor functions.

Abstract

The analysis of regularities and randomness in the distribution of prime numbers remains at the research frontiers for many generations of mathematicians from different groups and topical fields. In 2019 D. Fridman et al. (Am. Math. Mon. 2019, 126:1, 70-73) have suggested the constant $f_1 = 2.9200509773...$ for generation of the complete sequence of primes with using of a recursive relation for $f_n$ such that the floor function $\lfloor f_n \rfloor = p_n$, where $p_n$ is the nth prime. Here I present the family of constants $h_n (h_1 = 1.2148208055...)$ such that the ceiling function $\lceil h_n \rceil = p_n$. The proposed recursive relation $h_n=\lceil h_n \rceil(h_{n-1}-\lceil h_{n-1} \rceil+2)$ generates the sequence of all known prime numbers. I also show that constants $h_n$ are irrational.