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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.11282 |
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| _version_ | 1866917403009482752 |
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| author | Kominers, Scott Duke |
| author_facet | Kominers, Scott Duke |
| contents | We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of factorials. For a rational $\frac{r}{Q}$ with $\gcd(Q,m)=1$ and a fixed prime $p_0\nmid m$, membership in $K_{m,D}$ forces $ν_{p_0}(Q)$ to be controlled by the $p_0$-adic valuation of the multiplicative order of $m$ modulo the radical of $Q$, with explicit overhead depending only on $m$ and $D$. Because the obstruction is stated at the level of a single denominator, a pair of sequence-specific valuation estimates converts it into an effective finiteness criterion for $\left\{\frac{1}{a_n}:n\in\mathbb{N}\right\}\cap K_{m,D}$. Specializing to the case in which $Q$ is the part of $n!$ coprime to $m$ recovers the fixed-prime step in the Lin--Wu--Yang argument. As applications, we treat reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers. We also exhibit an exponential family -- products of $(m^k-1)$ -- for which the full structural criterion applies but a coarser largest-prime-factor formulation does not. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11282 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets Kominers, Scott Duke Number Theory Primary 11A63, Secondary 11A07 We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of factorials. For a rational $\frac{r}{Q}$ with $\gcd(Q,m)=1$ and a fixed prime $p_0\nmid m$, membership in $K_{m,D}$ forces $ν_{p_0}(Q)$ to be controlled by the $p_0$-adic valuation of the multiplicative order of $m$ modulo the radical of $Q$, with explicit overhead depending only on $m$ and $D$. Because the obstruction is stated at the level of a single denominator, a pair of sequence-specific valuation estimates converts it into an effective finiteness criterion for $\left\{\frac{1}{a_n}:n\in\mathbb{N}\right\}\cap K_{m,D}$. Specializing to the case in which $Q$ is the part of $n!$ coprime to $m$ recovers the fixed-prime step in the Lin--Wu--Yang argument. As applications, we treat reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers. We also exhibit an exponential family -- products of $(m^k-1)$ -- for which the full structural criterion applies but a coarser largest-prime-factor formulation does not. |
| title | A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets |
| topic | Number Theory Primary 11A63, Secondary 11A07 |
| url | https://arxiv.org/abs/2604.11282 |