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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2502.19427 |
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| _version_ | 1866929733757829120 |
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| author | Hirsh, Jordan |
| author_facet | Hirsh, Jordan |
| contents | Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = α_iα_{i-1}\dots α_0$ and $B = β_iβ_{i-1}\dots β_0$. Lucas proved that $$\binom{A}{B} \equiv \prod_{j=0}^{j=i}\binom{α_j}{β_j} \text{ mod } p.$$ Similarly as proved by Kummer, the $p$-adic valuation $v_p\binom{A}{B}$ is the number of borrows when computing $A-B$ in base $p$, or the number of carries in $(A-B)+B$ in base $p$. Davis and Webb discovered a generalization of Lucas's Theorem for prime powers. We prove a similar generalization in a different form using the concept of pseudo-digits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Lucas Theorem Hirsh, Jordan General Mathematics 11B65 (Primary), 05A10 (Secondary) Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = α_iα_{i-1}\dots α_0$ and $B = β_iβ_{i-1}\dots β_0$. Lucas proved that $$\binom{A}{B} \equiv \prod_{j=0}^{j=i}\binom{α_j}{β_j} \text{ mod } p.$$ Similarly as proved by Kummer, the $p$-adic valuation $v_p\binom{A}{B}$ is the number of borrows when computing $A-B$ in base $p$, or the number of carries in $(A-B)+B$ in base $p$. Davis and Webb discovered a generalization of Lucas's Theorem for prime powers. We prove a similar generalization in a different form using the concept of pseudo-digits. |
| title | Generalized Lucas Theorem |
| topic | General Mathematics 11B65 (Primary), 05A10 (Secondary) |
| url | https://arxiv.org/abs/2502.19427 |