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

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.