Some Diophantine Equations involving associated Pell numbers and repdigits

Gopal Krishna Panda; Monalisa Mohapatra; Pritam Kumar Bhoi

arxiv: 2411.00001 · v2 · pith:YBN5BW5Gnew · submitted 2024-10-08 · 🧮 math.GM

Some Diophantine Equations involving associated Pell numbers and repdigits

Monalisa Mohapatra , Pritam Kumar Bhoi , Gopal Krishna Panda This is my paper

Pith reviewed 2026-05-23 19:38 UTC · model grok-4.3

classification 🧮 math.GM

keywords Diophantine equationsassociated Pell numbersrepdigitslinear forms in logarithmsBaker-Davenport reductionconcatenationsdifferences

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The pith

All solutions are determined for repdigits as differences of associated Pell numbers, associated Pell numbers as differences of repdigits, and associated Pell numbers as concatenations of three repdigits.

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

The paper solves three families of Diophantine equations that connect repdigits with associated Pell numbers. It finds every case in which a repdigit equals the difference of two associated Pell numbers. It also finds every associated Pell number that equals the difference of two repdigits. Finally, it finds every associated Pell number that can be written as the concatenation of three repdigits. A reader cares because these results give an exhaustive classification of the intersections between two well-studied sequences under simple arithmetic and digit constraints.

Core claim

Using Baker's theory on linear forms in logarithms of algebraic numbers together with the Baker-Davenport reduction technique, and supported by direct computations in Mathematica, the authors determine all solutions to the three equation families: repdigits equal to the difference of two associated Pell numbers, associated Pell numbers equal to the difference of two repdigits, and associated Pell numbers that arise as the concatenation of three repdigits.

What carries the argument

Baker's theory on linear forms in logarithms combined with the Baker-Davenport reduction technique, which supplies effective upper bounds that reduce each equation to a finite computational search.

If this is right

The complete finite lists of solutions for each of the three equation types are now known.
No further solutions exist beyond those identified by the bounds and the program.
Associated Pell numbers satisfying the repdigit-difference or concatenation conditions are fully characterized.
The same bounding method applies directly to similar equations involving other linear recurrence sequences and restricted digit forms.

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 intersections of associated Pell numbers with other restricted digit patterns such as palindromes or numbers with few distinct digits.
Results of this type suggest that solutions to recurrence-sequence equations under digit constraints are sparse and can be reduced to finite checks in many cases.
One could test whether the listed solutions remain exhaustive when the associated Pell sequence is replaced by a generalized version with a different initial parameter.

Load-bearing premise

Baker's theory on linear forms in logarithms combined with the Baker-Davenport reduction provides effective bounds that, together with the Mathematica computations, exhaustively identify all solutions without omissions.

What would settle it

An explicit solution to any of the three equations that lies outside the derived bounds or was not returned by the Mathematica search program.

read the original abstract

In this paper, we explore the relationship between repdigits and associated Pell numbers, specifically focusing on two main aspects: expressing repdigits as the difference of two associated Pell numbers, and identifying which associated Pell numbers can be represented as the difference of two repdigits. Additionally, we investigate all associated Pell numbers which are the concatenation of three repdigits. Our proof utilizes Baker's theory on linear forms in logarithms of algebraic numbers, along with the Baker-Davenport reduction technique. The computations were carried out with the help of a simple computer program in {\it Mathematica}.

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

Standard Baker-Davenport application to associated Pell numbers and repdigits that fills in specific cases without new techniques.

read the letter

The main takeaway is that this paper takes the usual Baker theory plus Baker-Davenport reduction and applies it to three families of equations mixing associated Pell numbers with repdigits. It determines all solutions for repdigits as differences of those Pell numbers, the reverse, and associated Pell numbers formed by concatenating three repdigits. The work is a direct extension of an existing program rather than a shift in method or insight. The authors set up the linear forms, reduce the bounds, and finish with Mathematica checks, which matches what these papers normally do. That part is executed cleanly enough on the evidence of the abstract and method outline. The new element is simply the choice of associated Pell numbers instead of ordinary ones, plus the concatenation case, and the paper appears to carry the standard argument through without obvious setup errors. Credit is due for keeping the reduction effective and for using computation only after the bounds are brought down to a feasible range. The soft spots are limited. The abstract gives no explicit bounds or final solution list, so completeness rests on trusting the computation step, but that is routine and the description shows no sign of missed cases or incorrect linear forms. No circularity or invented entities appear. This paper is for number theorists who follow work on repdigits and linear recurrences; anyone already reading similar papers on Pell-like sequences will get the concrete results they expect. It will not interest a broader audience. I would send it to peer review because the approach is established, the claims are concrete and falsifiable in principle, and the subfield benefits from having these cases settled. An editor should let referees check the details rather than reject outright.

Referee Report

0 major / 2 minor

Summary. The manuscript determines all solutions to three families of Diophantine equations: repdigits expressed as differences of associated Pell numbers, associated Pell numbers expressed as differences of repdigits, and associated Pell numbers that are concatenations of three repdigits. The proofs rely on Baker's theory of linear forms in logarithms together with the Baker-Davenport reduction technique, supplemented by exhaustive Mathematica computations.

Significance. If the results hold, the paper supplies complete, explicit lists of solutions for these specific equations. This constitutes a standard but useful addition to the literature on exponential Diophantine equations involving linear recurrences and repdigits; the combination of effective bounds with machine verification is a recognized strength that supports exhaustiveness claims.

minor comments (2)

[Abstract] The abstract states that computations were performed with a 'simple computer program in Mathematica' but does not specify the range of the search or the precision used in the Baker-Davenport reduction; this detail belongs in §4 or an appendix.
[Section 2] Notation for associated Pell numbers (e.g., P_n^*) is introduced without an explicit recurrence or initial conditions in the preliminaries; a short definition paragraph would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The report provides a summary of the work but lists no specific major comments under the MAJOR COMMENTS section. Accordingly, we have no individual points to address point-by-point. We remain available to supply further details or clarifications should the editor or referee request them.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper determines solutions to Diophantine equations involving associated Pell numbers and repdigits by applying Baker's theory on linear forms in logarithms (an external, well-established result) together with the Baker-Davenport reduction technique and direct Mathematica verification. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, smuggled ansatzes, or renamings of known results appear in the derivation chain. The central claims rest on independent external theorems and exhaustive computational checks rather than reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper depends on standard results from Diophantine approximation without introducing new free parameters or entities.

axioms (1)

standard math Baker's theory on linear forms in logarithms of algebraic numbers provides effective bounds for solutions of the Diophantine equations considered
Invoked to prove finiteness and obtain explicit upper bounds on solutions

pith-pipeline@v0.9.0 · 5621 in / 1165 out tokens · 41814 ms · 2026-05-23T19:38:57.813975+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Bhoi and P

K. Bhoi and P. K. Ray, Repdigits as diﬀerence of two Fibonacci or Lucas numbers , Mat. Stud., 56 (2021), 124-132

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Bugeaud, M

Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular ap proaches to exponential Dio- phantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), 969-1018

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A. Dujella and A. Peth˝ o, A generalization of a theorem of Baker a nd Davenport, Quart. J. Math. Oxford Ser. 49 (1998), 291-306

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Pure Appl

MG Duman, Padovan numbers as diﬀerence of two repdigits, Indian J. Pure Appl. Math. https://doi.org/10.1007/s13226-023-00526-8, (2023)

work page doi:10.1007/s13226-023-00526-8 2023
[6]

Edjeou and B

B. Edjeou and B. Faye, Pell, and Pell–Lucas numbers as diﬀerence of two repdigits, Afr. Mat. 34 (2023), 70

work page 2023
[7]

Erduvan, R

F. Erduvan, R. Keskin, and F. Luca, Fibonacci, and Lucas numbe rs as the diﬀerence of two repdigits, Rend. Circ. Mat. Palermo, II. Ser 2 (2021), 1-5

work page 2021
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G´ uzman S´ anchez and F

S. G´ uzman S´ anchez and F. Luca,Linear combination of factorials and s-units in a binary rec urrence sequence, Ann. Math. Qu´ e,38 (2014), 169-188

work page 2014
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Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv

E.M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Math. 64 (2000), 1217-1269

work page 2000
[10]

Mohapatra, P

M. Mohapatra, P. K. Bhoi, and G. K. Panda, Repdigits as diﬀeren ce of two balancing or Lucas- balancing numbers, preprint, arXiv:2405.04801

work page arXiv
[11]

Mohapatra, P

M. Mohapatra, P. K. Bhoi, and G. K. Panda, Balancing and Lucas -balancing numbers as diﬀerence of two repdigits, preprint, arXiv:2405.15839

work page arXiv
[12]

S. G. Rayaguru and J. J. Bravo, Balancing and Lucas-balancing numbers which are concatenation of three repdigits, Bol. Soc. Mat. Mex. (3) 29 (2023), 57

work page 2023
[13]

S. G. Rayaguru and G. K. Panda, Balancing and Lucas-balancing numbers expressible as sums of two repdigits, Integers 21 (2021) p1, A7

work page 2021
[14]

S. G. Rayaguru and G. K. Panda, Repdigits as product of conse cutive balancing or Lucas-balancing numbers, Fibonacci Quart. 56(4) (2018), 319-324

work page 2018
[15]

S. G. Rayaguru and G. K. Panda and Z. S ¸iar, Associated Pell nu mbers which are repdigits or concatenation of two repdigits, Bol. Soc. Mat. Mex. (3). 27 (2021), 54. Monalisa Mohapatra, Department of Mathematics, National I nstitute of Technol- ogy Rourkela, Odisha-769 008, India Email address : mmahapatra0212@gmail.com Pritam Kumar Bhoi, Department of Ma...

work page 2021

[1] [1]

Bhoi and P

K. Bhoi and P. K. Ray, Repdigits as diﬀerence of two Fibonacci or Lucas numbers , Mat. Stud., 56 (2021), 124-132

work page 2021

[2] [2]

Bugeaud, M

Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular ap proaches to exponential Dio- phantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), 969-1018

work page 2006

[3] [3]

B. M. M. De Weger, Algorithms for diophantine equations , CWI Tracts 65. Stichting Mathematisch Centrum, Amsterdam, Centrum voor Wiskunde en Informatica (19 89)

work page

[4] [4]

Dujella and A

A. Dujella and A. Peth˝ o, A generalization of a theorem of Baker a nd Davenport, Quart. J. Math. Oxford Ser. 49 (1998), 291-306

work page 1998

[5] [5]

Pure Appl

MG Duman, Padovan numbers as diﬀerence of two repdigits, Indian J. Pure Appl. Math. https://doi.org/10.1007/s13226-023-00526-8, (2023)

work page doi:10.1007/s13226-023-00526-8 2023

[6] [6]

Edjeou and B

B. Edjeou and B. Faye, Pell, and Pell–Lucas numbers as diﬀerence of two repdigits, Afr. Mat. 34 (2023), 70

work page 2023

[7] [7]

Erduvan, R

F. Erduvan, R. Keskin, and F. Luca, Fibonacci, and Lucas numbe rs as the diﬀerence of two repdigits, Rend. Circ. Mat. Palermo, II. Ser 2 (2021), 1-5

work page 2021

[8] [8]

G´ uzman S´ anchez and F

S. G´ uzman S´ anchez and F. Luca,Linear combination of factorials and s-units in a binary rec urrence sequence, Ann. Math. Qu´ e,38 (2014), 169-188

work page 2014

[9] [9]

Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv

E.M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Math. 64 (2000), 1217-1269

work page 2000

[10] [10]

Mohapatra, P

M. Mohapatra, P. K. Bhoi, and G. K. Panda, Repdigits as diﬀeren ce of two balancing or Lucas- balancing numbers, preprint, arXiv:2405.04801

work page arXiv

[11] [11]

Mohapatra, P

M. Mohapatra, P. K. Bhoi, and G. K. Panda, Balancing and Lucas -balancing numbers as diﬀerence of two repdigits, preprint, arXiv:2405.15839

work page arXiv

[12] [12]

S. G. Rayaguru and J. J. Bravo, Balancing and Lucas-balancing numbers which are concatenation of three repdigits, Bol. Soc. Mat. Mex. (3) 29 (2023), 57

work page 2023

[13] [13]

S. G. Rayaguru and G. K. Panda, Balancing and Lucas-balancing numbers expressible as sums of two repdigits, Integers 21 (2021) p1, A7

work page 2021

[14] [14]

S. G. Rayaguru and G. K. Panda, Repdigits as product of conse cutive balancing or Lucas-balancing numbers, Fibonacci Quart. 56(4) (2018), 319-324

work page 2018

[15] [15]

S. G. Rayaguru and G. K. Panda and Z. S ¸iar, Associated Pell nu mbers which are repdigits or concatenation of two repdigits, Bol. Soc. Mat. Mex. (3). 27 (2021), 54. Monalisa Mohapatra, Department of Mathematics, National I nstitute of Technol- ogy Rourkela, Odisha-769 008, India Email address : mmahapatra0212@gmail.com Pritam Kumar Bhoi, Department of Ma...

work page 2021