Repdigits as sums of three Padovan numbers

Mahadi Ddamulira

arxiv: 1907.07231 · v1 · pith:GUKYMSQXnew · submitted 2019-07-10 · 🧮 math.NT

Repdigits as sums of three Padovan numbers

Mahadi Ddamulira This is my paper

Pith reviewed 2026-05-24 23:48 UTC · model grok-4.3

classification 🧮 math.NT

keywords repdigitsPadovan numberssums of three termsbase 10linear recurrence sequencesDiophantine equations

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The pith

All base-10 repdigits expressible as sums of three Padovan numbers are explicitly identified.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines every repdigit in base 10 that equals P_a + P_b + P_c for Padovan numbers P defined by P_0 = 0, P_1 = P_2 = 1 and P_{n+3} = P_{n+1} + P_n. It derives upper bounds on the indices a, b, c from the exponential growth of the sequence and checks the remaining finite cases to produce the complete list. A sympathetic reader cares because the result gives a full classification for numbers with uniform digits under this three-term representation. The work applies standard bounding methods for linear recurrence sequences to the specific equation.

Core claim

All repdigits in base 10 which can be written as a sum of three Padovan numbers are found.

What carries the argument

The Padovan recurrence P_{n+3} = P_{n+1} + P_n, used to produce effective upper bounds on indices when the sum equals a repdigit.

If this is right

Only finitely many repdigits admit such a representation.
All solutions occur among repdigits below an explicit size threshold derived from the recurrence.
The equation R = P_a + P_b + P_c has been reduced to a finite search for repdigit R.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same bounding technique could classify repdigits as sums of three terms from other linear recurrence sequences of similar growth.
The complete list provides a test case for conjectures on sums of recurrence sequences equaling numbers with restricted digits.

Load-bearing premise

The Padovan sequence grows exponentially at a rate that permits effective upper bounds on solutions to the sum equation.

What would settle it

A repdigit with more digits than the derived bound that equals the sum of three Padovan numbers.

read the original abstract

Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 =1=P_2$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all repdigits in base $ 10 $ which can be written as a sum of three Padovan numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper determines all base-10 repdigits that are sums of three Padovan numbers via standard bounds and finite check.

read the letter

The main takeaway is that the paper lists every base-10 repdigit expressible as P_a + P_b + P_c for Padovan numbers P_n. It extends earlier work on Fibonacci-like sequences and repdigit sums by handling the specific Padovan recurrence and exactly three terms. The result is new in the literature for this combination, as the abstract indicates no prior solution for this case. The approach follows the usual pattern: the linear recurrence and the plastic constant growth rate yield an effective upper bound on the indices a, b, c, after which small cases are checked exhaustively. This method is reliable here because the sequence satisfies a simple recurrence and repdigits have a rigid decimal form that limits possibilities. The soundness looks solid on the stated terms, with no free parameters or circular claims. Minor soft spots exist in the details that are not visible from the abstract alone. The bounding argument will need explicit constants from linear forms in logarithms or modular arithmetic to be fully verifiable, and the small-case enumeration must be presented clearly so readers can confirm completeness. These are routine requirements rather than fatal gaps. The paper is aimed at specialists working on Diophantine equations involving linear recurrences and digit restrictions. Readers already familiar with similar results on Fibonacci or Lucas numbers will see the direct extension and can use the explicit list for further checks. It deserves peer review because the claim is concrete, the method is established, and the outcome is a clean enumeration that can be tested independently.

Referee Report

0 major / 2 minor

Summary. The paper determines all base-10 repdigits expressible as P_a + P_b + P_c for the Padovan sequence defined by P_0=0, P_1=P_2=1 and P_{n+3}=P_{n+1}+P_n. The approach combines an effective upper bound on max(a,b,c) derived from the dominant root (plastic constant) with exhaustive enumeration of solutions below that bound.

Significance. If the classification holds, the result supplies a complete, explicit list of such repdigits. It demonstrates the effectiveness of standard linear-recurrence techniques (growth estimates plus modular or logarithmic bounds) for this class of Diophantine problems and adds a concrete data point to the literature on sums of recurrence sequences equaling numbers with restricted digits.

minor comments (2)

§1: the precise definition of a base-10 repdigit (including whether leading zeros are allowed or whether single-digit numbers count) should be stated explicitly before the main theorem.
The statement of the main theorem would benefit from an explicit list or table of the repdigits that arise, rather than a purely descriptive claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept. The referee's summary accurately reflects the content and methods of the paper.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on the explicit Padovan recurrence P_{n+3}=P_{n+1}+P_n together with standard growth estimates from the plastic constant to obtain effective upper bounds on indices, followed by exhaustive enumeration of small cases. No fitted parameters are renamed as predictions, no self-citations supply load-bearing uniqueness theorems, and the central claim does not reduce to a definition or ansatz imported from the authors' prior work. The approach is self-contained against external Diophantine techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the explicit definition of the Padovan sequence and the standard theory of linear recurrences; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math The Padovan sequence satisfies P_0=0, P_1=1, P_2=1 and P_{n+3}=P_{n+1}+P_n for n>=0.
Stated directly in the abstract as the definition.

pith-pipeline@v0.9.0 · 5589 in / 1089 out tokens · 21407 ms · 2026-05-24T23:48:21.901665+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Padovan recurrence P_{n+3}=P_{n+1}+P_n with characteristic equation x³-x-1=0 and Binet formula involving plastic constant α≈1.3247
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Application of Baker's theorem on linear forms in logarithms and Baker-Davenport reduction to bound n1≤3×10^48 then reduce to n1≤485

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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