Primes and almost primes between cubes
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911517138485248 |
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| author | Johnston, Daniel R. Sorenson, Jonathan P. Thomas, Simon N. Webster, Jonathan E. |
| author_facet | Johnston, Daniel R. Sorenson, Jonathan P. Thomas, Simon N. Webster, Jonathan E. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_15564 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Primes and almost primes between cubes Johnston, Daniel R. Sorenson, Jonathan P. Thomas, Simon N. Webster, Jonathan E. Number Theory 11N05, 11N36, 11Y11, 11Y16 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. |
| title | Primes and almost primes between cubes |
| topic | Number Theory 11N05, 11N36, 11Y11, 11Y16 |
| url | https://arxiv.org/abs/2601.15564 |