On Frobenius Numbers of Shifted Power Sequences
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915927889543168 |
|---|---|
| author | Liu, Feihu Xin, Guoce |
| author_facet | Liu, Feihu Xin, Guoce |
| contents | We resolve the open problem of characterizing the Frobenius number $g(A)$ for shifted square sequences $A = (a, a+1^2, \ldots, a+k^2)$, confirming a conjecture of Einstein et al. (2007). By combining a combinatorial reduction to an optimization problem with Lagrange's Four-Square Theorem and generating function techniques, we derive an explicit formula for $g(A)$: a piecewise quadratic polynomial in $a$, classified by residue classes modulo $k^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02722 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Frobenius Numbers of Shifted Power Sequences Liu, Feihu Xin, Guoce Combinatorics 11D07, 05A15, 11B75, 11D04 We resolve the open problem of characterizing the Frobenius number $g(A)$ for shifted square sequences $A = (a, a+1^2, \ldots, a+k^2)$, confirming a conjecture of Einstein et al. (2007). By combining a combinatorial reduction to an optimization problem with Lagrange's Four-Square Theorem and generating function techniques, we derive an explicit formula for $g(A)$: a piecewise quadratic polynomial in $a$, classified by residue classes modulo $k^2$. |
| title | On Frobenius Numbers of Shifted Power Sequences |
| topic | Combinatorics 11D07, 05A15, 11B75, 11D04 |
| url | https://arxiv.org/abs/2210.02722 |