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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.11269 |
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| _version_ | 1866915494225772544 |
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| author | Ma, Wanli |
| author_facet | Ma, Wanli |
| contents | We investigate arithmetic properties of the sequence b(n) = B_MN(n) mod M obtained from the base-M to base-N shift map B_MN.We prove that b(n) is ultimately periodic exactly when every prime divisor of M also divides N; in that case we bound (and, for prime powers, determine) the minimal period.When the condition fails, b(n) supplies new solutions to the Prouhet-Tarry-Escott problem.To analyze this situation we introduce a family of finite-difference identities and use them to evaluate two weighted multivariate polynomial sums, thereby extending identities that arise from the classical sum-of-digits function (N=1). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities Ma, Wanli Number Theory 11A63, 11P05, 11B83 We investigate arithmetic properties of the sequence b(n) = B_MN(n) mod M obtained from the base-M to base-N shift map B_MN.We prove that b(n) is ultimately periodic exactly when every prime divisor of M also divides N; in that case we bound (and, for prime powers, determine) the minimal period.When the condition fails, b(n) supplies new solutions to the Prouhet-Tarry-Escott problem.To analyze this situation we introduce a family of finite-difference identities and use them to evaluate two weighted multivariate polynomial sums, thereby extending identities that arise from the classical sum-of-digits function (N=1). |
| title | A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities |
| topic | Number Theory 11A63, 11P05, 11B83 |
| url | https://arxiv.org/abs/2509.11269 |