Unconditional Prime-Representing Functions, Following Mills

@article{Elsholtz2020UnconditionalPF,
title={Unconditional Prime-Representing Functions, Following Mills},
author={Christian Elsholtz},
journal={The American Mathematical Monthly},
year={2020},
volume={127},
pages={639 - 642}
}
• C. Elsholtz
• Published 2 April 2020
• Mathematics
• The American Mathematical Monthly
Abstract Mills proved that there exists a real constant A > 1 such that for all the values are prime numbers. No explicit value of A is known, but assuming the Riemann hypothesis one can choose Here we give a first unconditional variant: is prime, where can be computed to millions of digits. Similarly, is prime, with
4 Citations
A Couple of Transcendental Prime-Representing Constants
• J. L. Varona
• Mathematics
The American Mathematical Monthly
• 2021
Abstract It is well known that the arithmetic nature of Mills’ prime-representing constant is uncertain: we do not know if Mills’ constant is a rational or irrational number. In the case of other
Guided by the primes -- an exploration of very triangular numbers
• Mathematics
• 2021
We present a string of results concerning very triangular numbers, paralleling fundamental theorems established for prime numbers throughout the history of their study, from Euclid (∼300 BCE) to
Primes between consecutive powers
This paper updates two explicit estimates for primes between consecutive powers. We find at least one prime between n3 and (n+1)3 for all n ≥ exp(exp(32.9)), and at least one prime in (n296, (n +
Topological properties and algebraic independence of sets of prime‐representing constants
• Mathematics
Mathematika
• 2022
Let (ck)k=1∞$(c_k)_{k=1}^\infty$ be a sequence of positive integers. We investigate the set of A>1$A>1$ such that the integer part of Ac1⋯ck$A^{c_1\cdots c_k}$ is always a prime number for every

References

SHOWING 1-10 OF 18 REFERENCES
A Prime-Representing Constant
• Mathematics
Am. Math. Mon.
• 2019
A constant and a recursive relation are presented to define a sequence fn such that the floor of fn is the nth prime and this constant generates the complete sequence of primes.
Prime-representing functions
We construct prime-representing functions. In particular we show that there exist real numbers α > 1 such that ⌈α2n⌉ is prime for all n ∈ ℕ. Indeed the set consisting of such numbers α has the
Determining Mills' Constant and a Note on Honaker's Problem
• Mathematics
• 2005
In 1947 Mills proved that there exists a constant A such that bA 3 n c is a prime for every positive integer n. Determining A requires determining an efiective Hoheisel type result on the primes in
An Explicit Result for Primes Between Cubes
We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function
Functions which represent prime numbers
W. H. Mills [1ii has proved that there is a real number A so that [Al3] is a prime for every positive integer n. L. Kuipers [2] has extended this and proved that for any positive integer c _3 there
Primes in Short Intervals
• Mathematics
• 2004
Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x+H)−ψ(x), for 0≤x≤N, is approximately
On the selection of subsequences
The problem we shall discuss in the following is concerned with the possibility of extracting from a given sequence a subsequence whose terms obey some relatively simple mathematical law. As a
The new book of prime number records
1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C.