On a Generalisation of a Function of Ron Graham's
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913826011611136 |
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| author | Adenwalla, Sarosh |
| author_facet | Adenwalla, Sarosh |
| contents | Ron Graham introduced a function, $g(n)$, on the non-negative integers, in the 1986 Issue $3$ Problems column of \textit{Mathematical Magazine}: For each non-negative integer $n$, $g(n)$ is the least integer $s$ so that the integers $n + 1, n + 2, \ldots , s-1, s$ contain a subset of integers, the product of whose members with $n$ is a square. Recently, many results about $g(n)$ were proved in [Kagey and Rajesh, ArXiv:2410.04728, 2024] and they conjectured a characterization of which $n$ satisfied $g(n)=2n$. For $m\geq 2$, they also introduced generalizations of $g(n)$ to $m$-th powers to explore. In this paper, we prove their conjecture and provide some results about these generalisations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Generalisation of a Function of Ron Graham's Adenwalla, Sarosh Number Theory Combinatorics 11A05 Ron Graham introduced a function, $g(n)$, on the non-negative integers, in the 1986 Issue $3$ Problems column of \textit{Mathematical Magazine}: For each non-negative integer $n$, $g(n)$ is the least integer $s$ so that the integers $n + 1, n + 2, \ldots , s-1, s$ contain a subset of integers, the product of whose members with $n$ is a square. Recently, many results about $g(n)$ were proved in [Kagey and Rajesh, ArXiv:2410.04728, 2024] and they conjectured a characterization of which $n$ satisfied $g(n)=2n$. For $m\geq 2$, they also introduced generalizations of $g(n)$ to $m$-th powers to explore. In this paper, we prove their conjecture and provide some results about these generalisations. |
| title | On a Generalisation of a Function of Ron Graham's |
| topic | Number Theory Combinatorics 11A05 |
| url | https://arxiv.org/abs/2504.19196 |