On a Variant of Pillai's problem involving convergent denominators of quadratic irrationals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915447480254464 |
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| author | Mittal, Mohit |
| author_facet | Mittal, Mohit |
| contents | Let $(q_{α, n})_{n \geq 0}$ be the sequence of convergent denominators to the simple continued fraction expansion of $α$. For certain specific choices of $α$, this sequence is a Lehmer sequence. In this paper, we show that there are only finitely many integers $c$ such that the equation $q_{α, n} - q_{β, m} = c$ has at least two distinct solutions $(n,m)$, where $α,β$ are quadratic irrationals with $\mathbb{Q}(α)\neq \mathbb{Q}(β)$. In specific instances, we solve the equation $q_{α, n} - q_{β, m} = c$ completely and explicitly list all solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_11243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Variant of Pillai's problem involving convergent denominators of quadratic irrationals Mittal, Mohit Number Theory 11D45, 11D61, 11J86, 11R04, 11R11 Let $(q_{α, n})_{n \geq 0}$ be the sequence of convergent denominators to the simple continued fraction expansion of $α$. For certain specific choices of $α$, this sequence is a Lehmer sequence. In this paper, we show that there are only finitely many integers $c$ such that the equation $q_{α, n} - q_{β, m} = c$ has at least two distinct solutions $(n,m)$, where $α,β$ are quadratic irrationals with $\mathbb{Q}(α)\neq \mathbb{Q}(β)$. In specific instances, we solve the equation $q_{α, n} - q_{β, m} = c$ completely and explicitly list all solutions. |
| title | On a Variant of Pillai's problem involving convergent denominators of quadratic irrationals |
| topic | Number Theory 11D45, 11D61, 11J86, 11R04, 11R11 |
| url | https://arxiv.org/abs/2508.11243 |