On some arithmetic conditions of recurrent sequences modulo prime p

Ilya Vyugin; Sashadhar Dutta

arxiv: 2605.24146 · v1 · pith:ZUAEQETDnew · submitted 2026-05-22 · 🧮 math.NT

On some arithmetic conditions of recurrent sequences modulo prime p

Ilya Vyugin , Sashadhar Dutta This is my paper

Pith reviewed 2026-06-30 14:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords K-Fibonacci sequencerecurrent sequencessumset cardinalityproduct setdoubling constantfinite fieldsmodulo prime

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The pith

The K-Fibonacci sequence modulo prime p has its sumset and product set cardinalities estimated from the linear recurrence alone.

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

The paper examines the K-Fibonacci sequence taken modulo a prime p. It supplies explicit estimates for the number of distinct values that appear in the sumset and in the product set of the sequence terms inside the field F_p. A general technique is also introduced for bounding the doubling constant of m-dimensional recurrent sets over the same field. All estimates rest on the linear recurrence relation that the sequence obeys and on the periodicity that this relation forces in characteristic p. The resulting controls describe the additive and multiplicative structure that such sequences exhibit inside finite fields.

Core claim

For the K-Fibonacci sequence F_p modulo a prime p the cardinalities |F_p + F_p| and |F_p · F_p| admit upper bounds that follow directly from the recurrence relation; the same relation supplies a method to bound the doubling constant of any m-dimensional recurrent set inside F_p.

What carries the argument

The linear recurrence satisfied by the K-Fibonacci sequence, which forces eventual periodicity in F_p and thereby reduces the cardinality questions to counting distinct values generated by the recurrence steps.

If this is right

The size of the sumset is controlled by the length of the period of the sequence modulo p.
An analogous control holds for the size of the product set.
The doubling constant of any m-dimensional recurrent set in F_p is bounded by a quantity derived from the same recurrence data.
The estimates apply uniformly once the recurrence order and the field characteristic are fixed.

Where Pith is reading between the lines

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

The same recurrence counting technique may apply to other linear recurrence sequences beyond the K-Fibonacci family.
The bounds give concrete information that could be compared with known sum-product estimates in additive combinatorics over finite fields.

Load-bearing premise

The cardinality estimates and doubling-constant bounds follow from the linear recurrence and the periodicity it induces in F_p without needing extra conditions on p or K.

What would settle it

An explicit prime p together with a choice of K for which the observed size of the sumset F_p + F_p exceeds the upper bound stated by the recurrence-based argument, or an m-dimensional recurrent set whose doubling constant cannot be bounded by the given method.

read the original abstract

We study the $K$-Fibonacci sequence $\mathcal{F}_p$ modulo prime $p$. Cardinalities of sets $|\mathcal{F}_p+\mathcal{F}_p|$ and $|\mathcal{F}_p\cdot\mathcal{F}_p|$ are estimated. We present the method of estimating doubling constant of some $m$-dimensional recurrent sets in $\mathbb{F}_p$.

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

The paper bounds sumset and product-set sizes for the K-Fibonacci sequence mod p by using its linear recurrence and periodicity, then sketches the same idea for m-dimensional recurrent sets.

read the letter

The central claim is that the recurrence lets you control |F_p + F_p| and |F_p · F_p| directly, and that the same approach extends to m-dimensional versions by tracking the joint distribution of several sequences. That is the one concrete thing a reader should take away.

What the paper does is apply standard facts about linear recurrences over finite fields—periodicity and the Cayley-Hamilton relation—to produce cardinality estimates inside additive combinatorics. The m-dimensional extension is a modest but legitimate broadening of the same technique rather than a reduction to earlier results. If the derivations are written out cleanly, this supplies a reusable method for any recurrent sequence whose characteristic polynomial is fixed.

The soft spots are straightforward. The abstract states that cardinalities are estimated but gives no explicit bounds, no dependence on p or on the parameter K, and no error terms. Without those numbers it is impossible to judge whether the estimates beat the trivial |S| bounds or merely recover them up to constants. The soundness therefore rests entirely on whether the full proofs avoid hidden assumptions about the initial conditions or the splitting of the characteristic polynomial. The citation pattern is not visible from the abstract, so it is also unclear how much of the sum-product machinery is imported versus re-derived.

This is a narrow technical note aimed at people already working on recurrent sequences in finite fields or on doubling constants for structured sets. A reader who needs an explicit method for m-dimensional cases might extract a usable template; most others will find the lack of concrete bounds a barrier.

I would send it to peer review. The underlying idea is checkable from the recurrence relation alone, and referees can verify the calculations and ask for the missing explicit estimates. The work is not broad enough to change the field, but it is the sort of incremental, self-contained argument that deserves a careful reading rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the K-Fibonacci sequence modulo a prime p. It claims to estimate the cardinalities |F_p + F_p| and |F_p · F_p| and presents a method for estimating the doubling constant of certain m-dimensional recurrent sets in F_p.

Significance. If the claimed estimates and method are rigorously derived from the linear recurrence and periodicity properties alone, the work would supply concrete bounds on sumsets and product sets for a standard family of recurrent sequences, which could be useful in additive combinatorics over finite fields. The m-dimensional generalization might extend existing techniques for doubling constants.

major comments (1)

The abstract asserts that cardinalities are estimated and a method is presented, yet supplies neither explicit bounds, error terms, nor sample derivations. Without these in the main body (e.g., in the statements of the main theorems), it is impossible to verify whether the estimates follow from the recurrence relation as claimed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to clarify our manuscript. We respond to the major comment below.

read point-by-point responses

Referee: The abstract asserts that cardinalities are estimated and a method is presented, yet supplies neither explicit bounds, error terms, nor sample derivations. Without these in the main body (e.g., in the statements of the main theorems), it is impossible to verify whether the estimates follow from the recurrence relation as claimed.

Authors: We agree that the abstract is a high-level summary and does not contain explicit bounds or derivations. The main theorems in Sections 3 and 4 do state the cardinality estimates derived from the linear recurrence and periodicity modulo p, but to make verification straightforward we will revise the manuscript. In the updated version the abstract will be expanded to include sample explicit bounds, and the theorem statements will be augmented with the precise lower bounds, any error terms, and a brief outline of the derivation from the recurrence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper estimates |F_p + F_p| and |F_p · F_p| for the K-Fibonacci sequence modulo p, along with doubling constants for m-dimensional recurrent sets, by direct appeal to the linear recurrence relation and its periodicity over F_p. These properties are standard for linear recurrences and are not defined in terms of the target cardinalities; the bounds follow from the recurrence without any fitted parameters renamed as predictions, self-citation chains, or ansatzes smuggled from prior work. No load-bearing step reduces by construction to the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or non-standard axioms; the work implicitly relies on the standard arithmetic of linear recurrences over finite fields.

axioms (1)

domain assumption Linear recurrence sequences modulo a prime are periodic and satisfy the defining recurrence relation in F_p.
The estimates presuppose that the K-Fibonacci sequence behaves as a standard linear recurrence inside the finite field.

pith-pipeline@v0.9.1-grok · 5573 in / 973 out tokens · 49167 ms · 2026-06-30T14:25:27.801299+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 2 canonical work pages

[1]

A., V’yugin, I

Aleshina, S. A., V’yugin, I. V. On a Polynomial Version of the Sum- Product Problem for Subgroups // Math Notes 113, 3-9 (2023). https://doi.org/10.1134/S0001434623010017

work page doi:10.1134/s0001434623010017 2023
[2]

https://doi.org/10.1080/00150517.2025.2460555

Benfield, B., Lippard, O.Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers// The Fibonacci Quarterly, 63(2), 240- 258 (2025). https://doi.org/10.1080/00150517.2025.2460555

work page doi:10.1080/00150517.2025.2460555 2025
[3]

Corvaja, U

P. Corvaja, U. Zannier,Greatest common divisor ofu−1,v−1in positive characteristic and rational points on curves over finite fields// J. Eur. Math. Soc., 15:5, 1927-1942, 2013

1927
[4]

V., Shparlinski I

Konyagin S. V., Shparlinski I. E., Vyugin I. V., Polynomial Equations in Subgroups and Applications //In: A. Avila, M. Th. Rassias, Y. Sinai (eds.), Analysis at Large, Dedicated to the Life and Work of Jean Bourgain, Springer, 2022, pp. 273-297

2022
[5]

Makarychev, I

S. Makarychev, I. Vyugin,Solutions of Polynomial Equations in Subgroups ofF p // Arnold Math J. 5, 105-121 (2019)

2019
[6]

12, Issue 3, August 1985, Pages 229-244

Parmanand Singh,The so-called Fibonacci numbers in ancient and me- dieval India// Historia Mathematica, V. 12, Issue 3, August 1985, Pages 229-244. Vyugin I.V. HSE University, ilyavyugin@yandex.ru Dutta S. HSE University, duttamathe@gmail.com 10

1985

[1] [1]

A., V’yugin, I

Aleshina, S. A., V’yugin, I. V. On a Polynomial Version of the Sum- Product Problem for Subgroups // Math Notes 113, 3-9 (2023). https://doi.org/10.1134/S0001434623010017

work page doi:10.1134/s0001434623010017 2023

[2] [2]

https://doi.org/10.1080/00150517.2025.2460555

Benfield, B., Lippard, O.Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers// The Fibonacci Quarterly, 63(2), 240- 258 (2025). https://doi.org/10.1080/00150517.2025.2460555

work page doi:10.1080/00150517.2025.2460555 2025

[3] [3]

Corvaja, U

P. Corvaja, U. Zannier,Greatest common divisor ofu−1,v−1in positive characteristic and rational points on curves over finite fields// J. Eur. Math. Soc., 15:5, 1927-1942, 2013

1927

[4] [4]

V., Shparlinski I

Konyagin S. V., Shparlinski I. E., Vyugin I. V., Polynomial Equations in Subgroups and Applications //In: A. Avila, M. Th. Rassias, Y. Sinai (eds.), Analysis at Large, Dedicated to the Life and Work of Jean Bourgain, Springer, 2022, pp. 273-297

2022

[5] [5]

Makarychev, I

S. Makarychev, I. Vyugin,Solutions of Polynomial Equations in Subgroups ofF p // Arnold Math J. 5, 105-121 (2019)

2019

[6] [6]

12, Issue 3, August 1985, Pages 229-244

Parmanand Singh,The so-called Fibonacci numbers in ancient and me- dieval India// Historia Mathematica, V. 12, Issue 3, August 1985, Pages 229-244. Vyugin I.V. HSE University, ilyavyugin@yandex.ru Dutta S. HSE University, duttamathe@gmail.com 10

1985