End behavior of Ramanujan's taxicab numbers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913311271944192 |
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| author | Benfield, Brennan Lippard, Oliver Roy, Arindam |
| author_facet | Benfield, Brennan Lippard, Oliver Roy, Arindam |
| contents | Generalized taxicab numbers are the smallest positive integers that are the sum of exactly $j$, positive $k$-th powers in exactly $m$ distinct ways. This paper is considers for which values of $m$ does a smallest such integer exist as $j$ gets large. There appear to be only two possible outcomes, leading to curious results like there is no positive integer that can be expressed as the sum of exactly $10$ positive squares in exactly $3$ ways. This paper resolves a number of conjectures found in the OEIS by considering generalized Taxicab numbers in the setting of the theory of partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | End behavior of Ramanujan's taxicab numbers Benfield, Brennan Lippard, Oliver Roy, Arindam Number Theory Combinatorics 11P05, 11D09, 11D25, 11P81, 11Y50 Generalized taxicab numbers are the smallest positive integers that are the sum of exactly $j$, positive $k$-th powers in exactly $m$ distinct ways. This paper is considers for which values of $m$ does a smallest such integer exist as $j$ gets large. There appear to be only two possible outcomes, leading to curious results like there is no positive integer that can be expressed as the sum of exactly $10$ positive squares in exactly $3$ ways. This paper resolves a number of conjectures found in the OEIS by considering generalized Taxicab numbers in the setting of the theory of partitions. |
| title | End behavior of Ramanujan's taxicab numbers |
| topic | Number Theory Combinatorics 11P05, 11D09, 11D25, 11P81, 11Y50 |
| url | https://arxiv.org/abs/2404.08190 |