Diophantine tuples and product sets in shifted powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914394248577024 |
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| author | Croot, Ernie Yip, Chi Hoi |
| author_facet | Croot, Ernie Yip, Chi Hoi |
| contents | Let $k\geq 2$ and $n\neq 0$. A Diophantine tuple with property $D_k(n)$ is a set of positive integers $A$ such that $ab+n$ is a $k$-th power for all $a,b\in A$ with $a\neq b$. Such generalizations of classical Diophantine tuples have been studied extensively. In this paper, we prove several results related to robust versions of such Diophantine tuples and discuss their applications to product sets contained in a nontrivial shift of the set of all perfect powers or some of its special subsets. In particular, we substantially improve several results by Bérczes--Dujella--Hajdu--Luca, and Yip. We also prove several interesting conditional results. Our proofs are based on a novel combination of ideas from sieve methods, Diophantine approximation, and extremal graph theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04354 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine tuples and product sets in shifted powers Croot, Ernie Yip, Chi Hoi Number Theory Primary 11B30, 11D72, Secondary 11N36, 11D41, 05C35 Let $k\geq 2$ and $n\neq 0$. A Diophantine tuple with property $D_k(n)$ is a set of positive integers $A$ such that $ab+n$ is a $k$-th power for all $a,b\in A$ with $a\neq b$. Such generalizations of classical Diophantine tuples have been studied extensively. In this paper, we prove several results related to robust versions of such Diophantine tuples and discuss their applications to product sets contained in a nontrivial shift of the set of all perfect powers or some of its special subsets. In particular, we substantially improve several results by Bérczes--Dujella--Hajdu--Luca, and Yip. We also prove several interesting conditional results. Our proofs are based on a novel combination of ideas from sieve methods, Diophantine approximation, and extremal graph theory. |
| title | Diophantine tuples and product sets in shifted powers |
| topic | Number Theory Primary 11B30, 11D72, Secondary 11N36, 11D41, 05C35 |
| url | https://arxiv.org/abs/2504.04354 |