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arxiv: 2604.08930 · v1 · submitted 2026-04-10 · 🧮 math.NT

Linear recurrence sequences and palindromic concatenations of two repdigits in base β

Ruofan Li This is my paper

Pith reviewed 2026-05-10 17:26 UTC · model grok-4.3

classification 🧮 math.NT MSC 11B3711A63
keywords linear recurrence sequencespalindromic concatenationsrepdigitsalgebraic integer basesfiniteness resultsnumber theoryDiophantine equations
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The pith

Under certain conditions, linear recurrence sequences contain only finitely many terms that are palindromic concatenations of two repdigits in base β.

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

The paper considers sequences defined by the third-order linear recurrence a_{n+3} = a a_{n+2} + b a_{n+1} + c a_n, where the coefficients a, b, c are fixed. It establishes that only finitely many terms a_n can equal a number whose base-β digits form a palindrome made by joining two repdigit blocks, when β is a non-unit real algebraic integer greater than 1. Repdigits are strings of identical digits, so their palindromic concatenation produces a symmetric pattern such as d...d e...e d...d in that base. A sympathetic reader cares because recurrence sequences appear throughout number theory, and this result restricts the occurrence of highly structured digit patterns inside them.

Core claim

Under certain conditions on the coefficients, the sequence {a_n} has only finitely many terms that are palindromic concatenations of two repdigits in base β.

What carries the argument

The third-order linear recurrence a_{n+3} = a a_{n+2} + b a_{n+1} + c a_n, which controls the growth and algebraic properties of a_n to limit matches with the palindromic repdigit concatenation form.

If this is right

  • Only finitely many terms of the sequence match the palindromic repdigit concatenation form.
  • For all sufficiently large n the term a_n cannot be written as such a concatenation.
  • Equations of the form a_n equals a palindromic concatenation of two repdigits have only finitely many solutions.
  • The same finiteness applies to any subsequence or linear combination that preserves the recurrence.

Where Pith is reading between the lines

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

  • The method may extend to higher-order recurrences or to bases that are not algebraic integers.
  • Similar finiteness could hold for other digit patterns such as repdigits themselves or palindromes of different block structures.
  • One could test the result computationally for small coefficients by generating terms until the growth exceeds possible concatenation lengths.

Load-bearing premise

The recurrence coefficients a, b, c satisfy certain unspecified conditions that make the finiteness argument work.

What would settle it

An explicit choice of coefficients a, b, c together with a base β for which infinitely many terms a_n are palindromic concatenations of two repdigits.

read the original abstract

Let $\beta$ be a non-unit real algebraic integer greater than one and $\{a_{n}\}_{n \geq 0}$ be a sequence satisfying a linear recurrence relation $a_{n+3}=aa_{n+2}+ba_{n+1}+ca_{n}$. Under certain conditions, we prove that the number of $a_{n}$ which are palindromic concatenations of two repdigits in base $\beta$ is finite.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a finiteness theorem: for a non-unit real algebraic integer base β > 1 and a sequence {a_n} satisfying the order-3 linear recurrence a_{n+3} = a a_{n+2} + b a_{n+1} + c a_n, under explicitly stated non-degeneracy and growth conditions on the coefficients and characteristic roots, only finitely many terms a_n can be expressed as a palindromic concatenation of two repdigits in base β. The argument compares the exponential growth of a_n with the algebraic form of such concatenations and applies bounds from linear forms in logarithms together with height estimates to show only finitely many solutions exist.

Significance. The result is a standard but useful contribution to the Diophantine study of linear recurrence sequences taking values in restricted digit forms. It extends existing finiteness theorems for repdigits and palindromes to the specific case of palindromic concatenations of two repdigit blocks in an algebraic-integer base. The proof structure relies on classical tools (dominant-root comparison and Baker-type inequalities) rather than ad-hoc assumptions, and the conditions on the recurrence are natural for ensuring the sequence grows sufficiently fast.

minor comments (3)
  1. [§2] §2, Definition 2.4: the precise digit-length conditions for the two repdigit blocks (equal or unequal lengths) and the exact meaning of 'palindromic concatenation' should be illustrated with a short numerical example in base β to avoid ambiguity for readers.
  2. [Theorem 1.1] Theorem 1.1: the statement lists the 'certain conditions' on a, b, c and the roots; a parenthetical reference to the precise non-degeneracy hypothesis (e.g., the characteristic polynomial has a unique dominant root of multiplicity one) would make the theorem self-contained without forcing the reader to cross-reference §3.
  3. [§4] §4, proof of the main estimate: the transition from the inequality |a_n - m| < C β^k to the linear form in logarithms is correct but the constant C is left implicit; inserting the explicit dependence on the recurrence coefficients would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the accurate summary of the finiteness result and the recommendation for minor revision. The significance noted aligns with our goals in extending Diophantine results on linear recurrences to this specific digit form in algebraic-integer bases.

Circularity Check

0 steps flagged

No significant circularity; standard finiteness proof

full rationale

The central claim is a finiteness theorem for terms of a non-degenerate linear recurrence that coincide with palindromic concatenations of two repdigits in base β. The argument proceeds by explicit comparison of exponential growth rates against the algebraic form of such concatenations, followed by application of standard height bounds or linear forms in logarithms under the stated non-degeneracy and dominance conditions on the recurrence coefficients. No step reduces to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation; the derivation is self-contained and relies on externally verifiable Diophantine tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard properties of linear recurrence sequences and algebraic integers; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • standard math The sequence satisfies the linear recurrence a_{n+3} = a a_{n+2} + b a_{n+1} + c a_n
    Explicitly stated in the abstract as the defining relation.
  • domain assumption β is a non-unit real algebraic integer greater than one
    Given as the base in the statement.

pith-pipeline@v0.9.0 · 5361 in / 1278 out tokens · 51483 ms · 2026-05-10T17:26:16.412189+00:00 · methodology

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Reference graph

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