Primes and almost primes between cubes

TL;DR

This paper investigates prime and almost prime numbers between consecutive cubes, using large-scale computations and sieve methods to establish the existence of primes and semi-primes within these intervals up to very large bounds.

Contribution

It demonstrates the existence of primes and semi-primes between consecutive cubes for very large n, employing advanced computational and sieve techniques with numerical improvements.

Findings

Confirmed a prime exists between n^3 and (n+1)^3 for n^3 ≤ 1.649×10^40.

Proved a number with at most two prime factors exists in these intervals for all n ≥ 1.

Utilized a logarithmic weighting sieve method to improve previous results.

Abstract

In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition, we use this computation and a sieve-theoretic argument to show that there exists a number with at most 2 prime factors (counting multiplicity) between $n^3$ and $(n+1)^3$ for all $n\geq 1$. Our sieving argument uses a logarithmic weighting procedure attributed to Richert, which yields significant numerical improvements over previous approaches.