Sequences of integers generated by two fixed primes
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917105628086272 |
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| author | Languasco, Alessandro Luca, Florian Moree, Pieter Togbé, Alain |
| author_facet | Languasco, Alessandro Luca, Florian Moree, Pieter Togbé, Alain |
| contents | Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size $n_{i+1}-n_i$, with each bound containing an unspecified exponent and implicit constant. We will explicitly bound these four quantities. Earlier Langevin (1976) gave weaker estimates for (only) the exponents. Given a real number $α>1$, there exists a smallest number $m$ such that for every $n\ge m$, there exists an integer $n_i$ in $[n,nα)$. Our effective version of Tijdeman's result immediately implies an upper bound for $m$, which using the Koksma-Erdős-Turan inequality we will improve on. We present a fast algorithm to determine $m$ when $\max\{p,q\}$ is not too large and demonstrate it with numerical material. In an appendix we explain, given $n_i$, how to efficiently determine both $n_{i-1}$ and $n_{i+1}$, something closely related to work of Bérczes, Dujella and Hajdu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12806 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sequences of integers generated by two fixed primes Languasco, Alessandro Luca, Florian Moree, Pieter Togbé, Alain Number Theory 11B83, 11J70, 11J86, 11N25 Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size $n_{i+1}-n_i$, with each bound containing an unspecified exponent and implicit constant. We will explicitly bound these four quantities. Earlier Langevin (1976) gave weaker estimates for (only) the exponents. Given a real number $α>1$, there exists a smallest number $m$ such that for every $n\ge m$, there exists an integer $n_i$ in $[n,nα)$. Our effective version of Tijdeman's result immediately implies an upper bound for $m$, which using the Koksma-Erdős-Turan inequality we will improve on. We present a fast algorithm to determine $m$ when $\max\{p,q\}$ is not too large and demonstrate it with numerical material. In an appendix we explain, given $n_i$, how to efficiently determine both $n_{i-1}$ and $n_{i+1}$, something closely related to work of Bérczes, Dujella and Hajdu. |
| title | Sequences of integers generated by two fixed primes |
| topic | Number Theory 11B83, 11J70, 11J86, 11N25 |
| url | https://arxiv.org/abs/2309.12806 |