On the determination of $p$-Frobenius and related numbers using the $p$-Apéry set
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909071641149440 |
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| author | Komatsu, Takao |
| author_facet | Komatsu, Takao |
| contents | In this paper, we give convenient formulas in order to obtain explicit expressions of a generalized Frobenius number called the $p$-Frobenius number as well as its related values. Here, for a non-negative integer $p$, the $p$-Frobenius number is the largest integer whose number of solutions of the linear diophantine equation in terms of positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$ is at most $p$. When $p=0$, the problem is reduced to the famous and classical linear Diophantine problem of Frobenius. $0$-Frobenius number is the classical Frobenius number. Our formula is not only a natural extension of the existing classical formulas, but also has the great advantage that the explicit expressions of values such as the $p$-Frobenius and related numbers can be obtained systematically. The concept and formula of the weighted sum has been given recently. We also give a $p$-generalized formula for such weighted sums. The central role is the $p$-Apéry set, which is a generalization of the classical Apéry set. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_11021 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the determination of $p$-Frobenius and related numbers using the $p$-Apéry set Komatsu, Takao Number Theory Combinatorics 11D07, 05A15, 05A17, 05A19, 11B68, 11D04, 11P81, 20M14 In this paper, we give convenient formulas in order to obtain explicit expressions of a generalized Frobenius number called the $p$-Frobenius number as well as its related values. Here, for a non-negative integer $p$, the $p$-Frobenius number is the largest integer whose number of solutions of the linear diophantine equation in terms of positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$ is at most $p$. When $p=0$, the problem is reduced to the famous and classical linear Diophantine problem of Frobenius. $0$-Frobenius number is the classical Frobenius number. Our formula is not only a natural extension of the existing classical formulas, but also has the great advantage that the explicit expressions of values such as the $p$-Frobenius and related numbers can be obtained systematically. The concept and formula of the weighted sum has been given recently. We also give a $p$-generalized formula for such weighted sums. The central role is the $p$-Apéry set, which is a generalization of the classical Apéry set. |
| title | On the determination of $p$-Frobenius and related numbers using the $p$-Apéry set |
| topic | Number Theory Combinatorics 11D07, 05A15, 05A17, 05A19, 11B68, 11D04, 11P81, 20M14 |
| url | https://arxiv.org/abs/2111.11021 |