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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.19387 |
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| _version_ | 1866913673308536832 |
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| author | Thackeray, Henry Robert |
| author_facet | Thackeray, Henry Robert |
| contents | The exponential local-global principle, or Skolem conjecture, says: Suppose that \(b\) is a positive integer, and that the sequence \((u_{n})_{n = -\infty}^{\infty}\) is such that every term is in \(\mathbb{Z}[1/b]\), the linear recurrence \(u_{n + d} = a_{1}u_{n + d - 1} + \cdots + a_{d}u_{n}\) holds for all integers \(n\), and every root of \(x^{d} - a_{1}x^{d - 1} - a_{2}x^{d - 2} - \cdots - a_{d}\) is nonzero and simple; then there is no zero term \(u_{n}\) if and only if, for some integer \(m\) that is larger than \(1\) and relatively prime to \(b\), every term \(u_{n}\) is not in \(m\mathbb{Z}[1/b]\).
Particular cases of the conjecture are known, but the general conjecture is open. This paper proves some apparently new quadratic and degenerate cubic cases of the exponential local-global principle via power residue symbols.
This work was presented at the Stellenbosch Number Theory Conference 2025 in January 2025 at Stellenbosch University; much of the work was also presented at the 67th Annual Congress of the South African Mathematical Society in December 2024 at the University of Pretoria. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Power residue symbols and the exponential local-global principle Thackeray, Henry Robert Number Theory 11B37 The exponential local-global principle, or Skolem conjecture, says: Suppose that \(b\) is a positive integer, and that the sequence \((u_{n})_{n = -\infty}^{\infty}\) is such that every term is in \(\mathbb{Z}[1/b]\), the linear recurrence \(u_{n + d} = a_{1}u_{n + d - 1} + \cdots + a_{d}u_{n}\) holds for all integers \(n\), and every root of \(x^{d} - a_{1}x^{d - 1} - a_{2}x^{d - 2} - \cdots - a_{d}\) is nonzero and simple; then there is no zero term \(u_{n}\) if and only if, for some integer \(m\) that is larger than \(1\) and relatively prime to \(b\), every term \(u_{n}\) is not in \(m\mathbb{Z}[1/b]\). Particular cases of the conjecture are known, but the general conjecture is open. This paper proves some apparently new quadratic and degenerate cubic cases of the exponential local-global principle via power residue symbols. This work was presented at the Stellenbosch Number Theory Conference 2025 in January 2025 at Stellenbosch University; much of the work was also presented at the 67th Annual Congress of the South African Mathematical Society in December 2024 at the University of Pretoria. |
| title | Power residue symbols and the exponential local-global principle |
| topic | Number Theory 11B37 |
| url | https://arxiv.org/abs/2501.19387 |