Primes and almost primes between cubes

TL;DR

The paper tackles the existence of primes between consecutive cubes and strengthens almost-prime results in the same cubic intervals. It combines a large-scale computation that certifies a prime in every interval $(n^3,(n+1)^3)$ up to $n^3\le 1.649\cdot 10^{40}$ with an explicit sieve-theoretic framework built around Richert's logarithmic weights and an explicit linear sieve. This yields, for all $n\ge 1$, the existence of an $a\in(n^3,(n+1)^3)$ with $\Omega(a)\le 2$, improving prior bounds and clarifying the numerical efficiency of Richert's weighting over Kuhn's. The work also discusses limitations (non-reachability of $\Omega(a)=1$ with current methods) and potential extensions to higher powers, as well as a Mill's-prime-representing-function corollary.

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.

Theorems & Definitions (26)

• Theorem 1.1: Ingham
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 2.1: sorenson2025
• Theorem 2.2
• Lemma 3.1: Explicit version of the linear sieve BJV24
• Lemma 3.2
• Lemma 3.3: rosser1962approximate