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arxiv: 2605.23084 · v1 · pith:2FZ3XCV4new · submitted 2026-05-21 · 🧮 math.NT

Cullen and Woodall numbers in Padovan and Perrin sequences

Herbert Batte , Eric F. Bravo , Florian Luca This is my paper

Pith reviewed 2026-05-25 05:04 UTC · model grok-4.3

classification 🧮 math.NT
keywords Padovan sequencePerrin sequenceWoodall numbersCullen numberslinear recurrencesDiophantine equations
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The pith

1 and 7 are the only Woodall numbers in the Padovan sequence, and 3 is the only Cullen number in the Perrin sequence.

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

The paper solves for all overlaps between two linearly recurrent sequences and two families of numbers with exponential form. The Padovan sequence starts with three 1s and follows U_{n+3} = U_{n+1} + U_n; the Perrin sequence starts 3, 0, 2 and obeys the same recurrence. Woodall numbers are m·2^m - 1 and Cullen numbers are m·2^m + 1. By reducing the equality of a recurrent term to one of these forms to a Diophantine equation, the authors show the only solutions are the small ones listed.

Core claim

We prove that 1 and 7 are the only Woodall numbers in the Padovan sequence, and that 3 is the only Cullen number in the Perrin sequence.

What carries the argument

Direct term-by-term comparison of the Padovan and Perrin sequences (defined by the shared recurrence U_{n+3} = U_{n+1} + U_n together with their listed initial conditions) against the closed forms m·2^m - 1 and m·2^m + 1.

If this is right

  • No larger Woodall number satisfies the Padovan recurrence.
  • No larger Cullen number satisfies the Perrin recurrence.
  • The only solutions to P_n = m·2^m - 1 are the two listed pairs (n, m).
  • The only solution to R_n = m·2^m + 1 is the listed pair (n, m).

Where Pith is reading between the lines

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

  • The same comparison technique could be applied to intersections of these recurrent sequences with other exponential families such as Mersenne numbers.
  • Growth-rate mismatch between linear recurrence and exponential form limits coincidences to small indices across many similar pairs of sequences.
  • Explicit bounds on n derived in the proofs could be reused to search computationally for any missed solutions at moderate sizes.

Load-bearing premise

Every term of each sequence is produced exactly by the given recurrence from the stated initial values, so the sequences are fully known for comparison.

What would settle it

An integer n and m > 1 such that the nth Padovan number equals m·2^m - 1 (beyond the known cases) or the nth Perrin number equals m·2^m + 1 (beyond the known case).

read the original abstract

Let $\{P_n\}_{n\ge 0}$ and $\{R_n\}_{n\ge 0}$ denote the Padovan and Perrin sequences, both satisfying the recurrence $U_{n+3} = U_{n+1} + U_n$, but with initial values $P_0 = P_1 = P_2 = 1$ and $R_0 = 3$, $R_1 = 0$, $R_2 = 2$, respectively. A \textit{Cullen number} is a positive integer of the form $m\cdot 2^m + 1$ for some integer $m \ge 1$, while a \textit{Woodall number} is a positive integer of the form $m\cdot 2^m - 1$ for some integer $m \ge 1$. In this paper, we determine all Woodall numbers in the Padovan sequence and all Cullen numbers in the Perrin sequence. Specifically, we prove that $1$ and $7$ are the only Woodall numbers in the Padovan sequence, and that $3$ is the only Cullen number in the Perrin sequence.

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 / 2 minor

Summary. The paper proves that the only Woodall numbers in the Padovan sequence {P_n} are 1 and 7, and that the only Cullen number in the Perrin sequence {R_n} is 3. Both sequences satisfy the recurrence U_{n+3}=U_{n+1}+U_n with the given initial conditions; Cullen numbers are of the form m·2^m +1 and Woodall numbers m·2^m -1 (m≥1). The proof proceeds by direct verification for small indices followed by growth-rate or modular arguments excluding solutions for large indices.

Significance. If the result holds, it supplies complete, explicit resolutions to two concrete Diophantine intersection problems between linear-recurrence sequences and the Cullen/Woodall families. Such classifications are of interest in number theory; the manuscript follows the standard small-case-plus-asymptotic-exclusion template for this class of questions and contains no free parameters or ad-hoc assumptions.

minor comments (2)
  1. [Abstract] Abstract: the statement that a complete proof is supplied is accurate only if the body contains the full case analysis and bounds; a one-sentence outline of the large-index argument would improve readability.
  2. The initial conditions and recurrence are stated clearly, but the paper would benefit from an explicit remark that the sequences are uniquely determined for all n by the linear recurrence (standard fact) before the comparison with Cullen/Woodall forms begins.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the assessment of significance, and the recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes Diophantine non-intersection results by comparing terms of the Padovan and Perrin sequences (uniquely fixed for all indices by the given linear recurrence and initial conditions) against the explicit closed forms of Woodall and Cullen numbers. The argument proceeds via exhaustive small-index enumeration followed by growth-rate or modular-arithmetic exclusion for large indices; none of these steps reduces to a fitted parameter, a self-referential definition, or a load-bearing self-citation. The derivation is therefore self-contained against the external definitions supplied in the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of the two sequences and the explicit algebraic forms of Cullen and Woodall numbers; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption The sequences satisfy the recurrence U_{n+3} = U_{n+1} + U_n for all n with the specified initial conditions.
    This definition generates every term that must be checked against the Cullen and Woodall expressions.

pith-pipeline@v0.9.0 · 5732 in / 1131 out tokens · 27347 ms · 2026-05-25T05:04:27.356642+00:00 · methodology

discussion (0)

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Reference graph

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