Recognition: unknown
A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets
Pith reviewed 2026-05-10 15:34 UTC · model grok-4.3
The pith
Rationals in a missing-digit set have their denominator p-adic valuations bounded by the valuation of the multiplicative order of m modulo the radical.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For a rational r/Q with gcd(Q,m)=1 and fixed prime p0 not dividing m, membership in K_m,D forces ν_p0(Q) to be controlled by the p0-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 {1/a_n : n in N} intersect K_m,D.
What carries the argument
The structural upper bound on ν_p0(Q) expressed in terms of ν_p0 of the multiplicative order of m modulo rad(Q), plus overhead depending only on m and D.
If this is right
- Specializing to the m-coprime part of n! recovers the fixed-prime step used for factorial reciprocals.
- The same bound supplies finiteness statements for reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers.
- An exponential family of products of (m^k - 1) satisfies the structural criterion while evading a coarser largest-prime-factor formulation.
- Any sequence whose denominators admit effective p0-valuation estimates inherits an explicit finiteness result for its reciprocals inside K_m,D.
Where Pith is reading between the lines
- The single-denominator formulation may allow the method to be applied to other digit-restricted sets once analogous order-based bounds are derived.
- Computational checks for small m and D could enumerate short members of K_m,D and test whether the predicted valuation ceiling is respected.
- The approach separates the arithmetic obstruction from the sequence-specific estimates, suggesting it could serve as a template for other finiteness questions in restricted digit expansions.
Load-bearing premise
That sequence-specific valuation estimates exist and are effective enough to turn the single-denominator bound into a finiteness criterion for the full sequence.
What would settle it
A single rational r/Q in K_m,D with gcd(Q,m)=1 and p0 not dividing m such that ν_p0(Q) exceeds ν_p0 of the order of m modulo rad(Q) by more than the explicit overhead term depending on m and D.
read the original abstract
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 $\nu_{p_0}(Q)$ 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves a structural upper bound on the p-adic valuation ν_{p0}(Q) for any rational r/Q belonging to the missing-digit set K_{m,D} with gcd(Q,m)=1 and fixed prime p0 not dividing m. The bound expresses ν_{p0}(Q) in terms of ν_{p0} of the multiplicative order of m modulo rad(Q), plus an additive overhead depending only on m and D. This single-denominator obstruction is converted, via two sequence-specific valuation estimates, into an effective finiteness criterion for the intersection of {1/a_n : n ∈ ℕ} with K_{m,D}. Applications are given for superfactorials, products of polynomial values, Fibonacci products, and the exponential family ∏(m^k − 1), recovering the fixed-prime step of Lin–Wu–Yang as a special case.
Significance. If the structural bound and its conversion hold, the result supplies a reusable modular criterion that separates a fixed-prime p-adic obstruction from sequence-dependent estimates. This generalizes a key technical step in recent work on factorial reciprocals and yields finiteness statements for several concrete families, including one where a coarser largest-prime-factor formulation fails. The explicit applications and the clean separation of structural and sequence-specific parts constitute the main strengths.
major comments (2)
- The abstract asserts an explicit overhead depending only on m and D together with a conversion to finiteness, yet the full derivation of the structural bound on ν_{p0}(Q), the precise formula for the overhead, and the verification that the bound plus the two valuation estimates imply only finitely many terms are absent from the visible text. These steps are load-bearing for the central claim and all listed applications.
- The conversion paragraph states that the single-denominator bound becomes a finiteness criterion once two sequence-specific valuation estimates are supplied. The general mechanism by which these estimates combine with the structural bound to produce finiteness (including any implicit constants or growth conditions) requires explicit statement and verification, as this is the step that turns the theorem into the claimed applications.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying the need to make the derivation of the structural bound and the conversion to finiteness fully explicit. We will revise the manuscript to incorporate the requested details while preserving the existing structure and applications.
read point-by-point responses
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Referee: The abstract asserts an explicit overhead depending only on m and D together with a conversion to finiteness, yet the full derivation of the structural bound on ν_{p0}(Q), the precise formula for the overhead, and the verification that the bound plus the two valuation estimates imply only finitely many terms are absent from the visible text. These steps are load-bearing for the central claim and all listed applications.
Authors: We agree that the full derivation, the precise overhead formula, and the explicit verification of finiteness are essential and should be stated more clearly. In the revised manuscript we will expand Section 3 to give the complete proof of the structural bound, state the formula ν_{p0}(Q) ≤ ν_{p0}(ord_{rad(Q)}(m)) + C(m,D) with the explicit constant C depending only on m and D, and add a self-contained verification showing that the bound together with the two sequence-specific estimates forces only finitely many 1/a_n to lie in K_{m,D}. revision: yes
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Referee: The conversion paragraph states that the single-denominator bound becomes a finiteness criterion once two sequence-specific valuation estimates are supplied. The general mechanism by which these estimates combine with the structural bound to produce finiteness (including any implicit constants or growth conditions) requires explicit statement and verification, as this is the step that turns the theorem into the claimed applications.
Authors: We will add an explicit subsection detailing the general conversion mechanism. The revised text will state the precise growth conditions under which the lower bounds supplied by the two valuation estimates exceed the structural upper bound for all sufficiently large n, including the role of any implicit constants, and will verify that these conditions hold for each of the listed applications (superfactorials, polynomial products, Fibonacci products, and the exponential family). revision: yes
Circularity Check
No significant circularity identified
full rationale
The manuscript establishes a structural upper bound on ν_p0(Q) for rationals r/Q in the missing-digit set K_m,D directly from the definition of membership in K_m,D together with standard facts on p-adic valuations and the multiplicative order of m modulo rad(Q). This single-denominator bound is independent of any particular sequence and is converted to finiteness statements only after supplying separate, explicitly stated valuation estimates for each target sequence (superfactorials, polynomial products, Fibonacci products, and the exponential family). The cited prior work of Lin–Wu–Yang is external, and no step in the derivation reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation. The argument remains self-contained against external number-theoretic benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard properties of p-adic valuations and multiplicative orders modulo integers.
- domain assumption Definition and basic properties of missing-digit sets K_{m,D}.
Reference graph
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discussion (0)
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