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arXiv · 2601.15564

Primes and almost primes between cubes

Daniel R. Johnston·Jonathan P. Sorenson·Simon N. Thomas·Jonathan E. Webster
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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.

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Daniel R. Johnston, Jonathan P. Sorenson, Simon N. Thomas, Jonathan E. Webster. 2026-01-22. Primes and almost primes between cubes. https://arxiv.org/abs/2601.15564

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