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10.12688/f1000research.181567.1

Research Article

Articles

Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers

[version 1; peer review: 2 not approved]

Derso

Derebew

Conceptualization Formal Analysis Investigation Methodology Project Administration Resources Supervision Validation Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0003-2431-2802 1 ADMASU

AGEZE

a ageze.ab19@gmail.com

No competing interests were disclosed.

14 5 2026

2026

734

28 4 2026

2026

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background

Linear recurrence sequences have been extensively studied in number theory and combinatory, with the Fibonacci sequence being the most classical example. Recent research has expanded to include various generalizations such as k-Fibonacci sequences ¹ Cullen sequences ² and polynomial extensions [see ^{3,
4}]. Among these, Mulatu numbers, introduced by Mulatu Lemma ⁵ and defined by the recurrence: M n = M n − 1 + M n − 2 , M 0 = 4 , M 1 = 1 , have emerged as an interesting variant with unique arithmetic properties. Recent work by Derso and Admasu ⁶ established several characterizations of Mulatu numbers, including sum formulas, divisibility properties, and connections to the golden ratio.

Methods

we develop and analyze an efficient detection algorithm for determining whether a given integer belongs to the Mulatu sequence, based on a perfect-square criterion and modular arithmetic. Our results unify and extend recent work on generalized by Fibonacci sequences and Lucas Sequences and provide new computational tools for number theory and discrete mathematics.

Results

We derive explicit Binet-type formulas, generating functions, and combinatorial identities, establishing deep connections with Fibonacci polynomials, Lucas’s polynomials, and other linear recurrence sequences.

Conclusions

This paper gives a polynomial generalization of Mulatu numbers that extends the classical recurrence M n = M n − 1 + M n − 2 to polynomial sequences.

Mulatu sequence Mulatu recurrence characteristic Mulatu polynomial Mulatu series.

The author(s) declared that no grants were involved in supporting this work.

1. Introduction

Linear recurrence sequences have been extensively studied in number theory and combinatorics, with the Fibonacci sequence being the most classical example. Recent research has expanded to include various generalizations such as k-Fibonacci sequences, ¹ Cullen sequences ² and polynomial extensions [see ^{3,
4}]. Among these, Mulatu numbers, introduced by Mulatu Lemma ⁵ and defined by the recurrence: M n = M n − 1 + M n − 2 , M 0 = 4 ; M 1 = 1 , have emerged as an interesting variant with unique arithmetic properties. Recent work by Derso and Admasu ⁶ established several characterizations of Mulatu numbers, including sum formulas, divisibility properties, and connections to the golden ratio.

However, two significant gaps remain in the literature. First, while polynomial generalizations exist for Fibonacci and Lucas sequences [see ^{7,
8}], no such extension has been developed for Mulatu numbers. Second, unlike for the Mulatu numbers, detection algorithms have done for the Fibonacci numbers ⁹ and Lucas numbers. ¹⁰

In this paper, we address both gaps by: 1.

Introducing Mulatu polynomials M n ( x ) and establishing their fundamental properties, including explicit formulas, generating functions, and combinatorial identities.

Deriving explicit connections between Mulatu polynomials and established polynomial sequences, particularly Fibonacci polynomials F n ( x ) and Lucas polynomials L n ( x ).

Developing a deterministic O (log N) algorithm for Mulatu number detection, proving its correctness via a perfect-square criterion and analyzing its computational complexity.

Proving new theorems that deepen the understanding of Mulatu sequences in both numeric and polynomial forms, connecting to recent work on generalized Fibonacci sequences.

Our work builds upon recent developments in special sequences [see ^{11,
12}] and computational number theory ¹³ contributing to both theoretical understanding and practical applications.

1.1 Literature review

The study of generalized Fibonacci sequences dates back to Horadam ¹⁴ and has seen renewed interest in recent years. The k-Fibonacci numbers ¹ and generalized Fibonacci polynomials ⁴ have applications in combinatorics, cryptography, and optimization. Mulatu numbers represent a specific case with initial conditions 4, 1, which distinguishes them from classical Fibonacci 0,1 and Lucas 2,1 sequences.

Recent work by Özkan and Akkuş ³ introduced the copper ratio through Fibonacci generalizations, while Akkuş et al. ² explored self-similarity in k-Cullen sequences. Adédji et al. ¹¹ investigated Mulatu numbers as products of three generalized Lucas numbers, demonstrating their rich arithmetic structure.

Fibonacci polynomials F n ( x ) and Lucas’s polynomials L n ( x ) have been extensively studied [see ^{7,
8}]. These satisfy: F n ( x ) = x F n − 1 ( x ) + F n − 2 ( x ) , F 0 ( x ) = 0 , F 1 ( x ) = 1 , L n ( x ) = x L n − 1 ( x ) + L n − 2 ( x ) , L 0 ( x ) = 2 , L 1 ( x ) = 1 .

Their properties include explicit Binet formulas, generating functions, and combinatorial interpretations. ¹⁵ Our extension of Mulatu numbers to polynomials follows this established framework while introducing new identities specific to the 4,1 initial conditions.

2. Preliminaries

The Mulatu numbers M n , introduced by Lemma, ¹⁶ form a second-order linear recurrence sequence defined by M n = M n − 1 + M n − 2 with initial conditions M 0 = 4 and M 1 = 1 , yielding the terms 4, 1, 5, 6, 11, 17, 28, 45, … . A foundational result in the literature is the explicit relationship between Mulatu numbers and the classical Fibonacci numbers F n (defined by F 0 = 0 , F 1 = 1 ) and Lucas numbers L n (defined by L 0 = 2 , L 1 = 1 ). Lemma and Lambright ¹⁷ established several identities connecting these sequences, demonstrating that Mulatu numbers can be expressed as linear combinations of Fibonacci and Lucas numbers. Specifically, the relationship M n = 4 F n − 1 + F n for n ≥ 1 and connections of the form M n = F n − 1 + L n have been derived. ¹⁷ Additional identities exploring the fascinating mathematical patterns among these three interrelated sequences have been extensively documented, revealing that the Mulatu numbers share the same recurrence relation as Fibonacci and Lucas numbers while exhibiting unique properties due to their distinct initial values. ^{16,
17}

The study of polynomial generalizations of these number sequences has a rich history. Fibonacci polynomials F n ( x ) and Lucas polynomials L n ( x ) are defined by the recurrence P n ( x ) = x P n − 1 ( x ) + P n − 2 ( x ) with initial conditions F 0 ( x ) = 0 , F 1 ( x ) = 1 for Fibonacci polynomials, and L 0 ( x ) = 2 , L 1 ( x ) = x for Lucas polynomials. ^{18,
19} These polynomial sequences reduce to the classical Fibonacci and Lucas numbers when evaluated at x = 1 , and to Pell numbers when evaluated at x = 2 . ¹⁹ Galvez and Dehesa ¹⁹ investigated novel properties of these polynomials, including the distribution of their zeros and evaluations of partial sums. Filipponi and Horadam ¹³ extended this work by studying the second derivative sequences of Fibonacci and Lucas polynomials, denoted F n ( 2 ) and L n ( 2 ) , deriving numerous identities such as F ( 2 ) ( n + m ) + ( − 1 ) m F ( 2 ) ( n − m ) = L m F n ( 2 ) + F n L n ( 2 ) + 2 m F m F n ( 1 ) . The closed-form Binet-type expressions for these polynomials are given by F n ( x ) = ( α ( x ) n − β ( x ) n ) / ( α ( x ) − β ( x ) ) and L n ( x ) = α ( x ) n + β ( x ) n , where α ( x ) , β ( x ) = x ± x 2 + 4 2 . ^{22,
23} These foundational results on Fibonacci and Lucas polynomials provide a natural framework for defining and investigating Mulatu polynomials as a new polynomial family with initial conditions M 0 ( x ) = 4 and M 1 ( x ) = 1 , extending the existing relationships between the classical number sequences to the polynomial domain.

3. Main results 3.1 Detection algorithms for special numbers

The problem of determining whether a given integer belongs to a specific sequence has practical applications in cryptography and coding theory. For Fibonacci numbers, efficient algorithms exist based on the identity that N is Fibonacci if and only if 5 N 2 ± 4 is a perfect square. ⁹ Similar criteria exist for Pell numbers and other second-order recurrences. ¹⁰ Our detection algorithm for Mulatu numbers builds upon these approaches while addressing the specific challenges posed by the 4,1 initial conditions.

Mulatu Polynomials: Definition and Fundamental Properties

Definition and Recurrence

Building on the definition of Mulatu numbers, ^{5,
6} we introduce Mulatu polynomials as follows:

The Mulatu polynomials M n ( x ) n = 0 ∞ are defined by the recurrence: M n ( x ) = x M n − 1 ( x ) + M n − 2 ( x ) , M 0 ( x ) = 4 , M 1 ( x ) = 1 .

For x = 1, we recover the classical Mulatu numbers: M n ( 1 ) = M n .

This definition follows the pattern of Fibonacci polynomials ⁷ and Lucas polynomials, ⁸ extending the numeric recurrence to a polynomial sequence.

Binet Formula for Mulatu Polynomials

Let α ( x ) = x + x 2 + 4 2 and β ( x ) = x − x 2 + 4 2 be the roots of the characteristic equation.

t 2 − xt − 1 = 0 . Then for all n ≥ 0 , M n ( x ) = ( 4 − β ( x ) ) α ( x ) n − ( 4 − α ( x ) ) β ( x ) n α ( x ) − β ( x ) .

The proof follows the standard method for solving linear recurrences with constant coefficients in the polynomial setting. ⁸ Substituting M n ( x ) = r n into the recurrence gives r 2 − xr − 1 = 0 , with roots α ( x ) and β ( x ) , the coefficients are determined by the initial conditions M 0 ( x ) = 4 and M 1 ( x ) = 1 .

This result generalizes the Binet formula for Mulatu numbers given in ²⁰ and parallels the formulas for Fibonacci and Lucas polynomials. ⁷

Generating Function: The generating function for Mulatu polynomials is: G ( x , t ) = ∑ n = 0 ∞ M n ( x ) t n = 4 − 3 t 1 − xt − t 2

Following the method for linear recurrences, ²¹ multiply the recurrence by t n , sum from n = 2 to ∞ , and solve for G ( x , t ) using the initial conditions.

This generating function resembles but differs from those for Fibonacci polynomials t 1 − xt − t 2

and Lucas’s polynomials 2 − xt 1 − xt − t 2 , ⁸ reflecting the different initial conditions.

Connections with Fibonacci and Lucas Polynomials

Explicit Relationships

Building on known relationships between numeric sequences, ¹⁶ we establish polynomial identities: Theorem:

For all n ≥ 1 , M n ( x ) = 4 F n − 1 ( x ) + F n ( x ) , where F n ( x ) denote Fibonacci polynomials.

Proof:

By induction. Assuming the identity holds for n = k and n = k − 1 , we have: M k + 1 ( x ) = x M k ( x ) + M k − 1 ( x ) = x [ 4 F k − 1 ( x ) + F k ( x ) ] + [ 4 F k − 2 ( x ) + F k − 1 ( x ) ] = 4 ( x F k − 1 + F k − 2 ) + ( x F k + F k − 1 ) = 4 F k ( x ) + F k + 1 ( x ) ,

Using the Fibonacci polynomial recurrence F k + 1 ( x ) = x F k ( x ) + F k − 1 ( x ) .

This theorem extends the numeric identity M n = 4 F n − 1 + F n from ⁶ to the polynomial setting.

Corollary:

For n ≥ 2 , M n ( x ) = L n ( x ) + 2 F n − 1 ( x ) , where L n ( x ) denote Lucas’s polynomials.

Proof:

From Theorem above and the identity L n ( x ) = F n − 1 ( x ) + F n + 1 ( x ) ⁸

Special Cases and Applications Proposition:

For x = 1 : M n ( 1 ) = M n , the classical Mulatu numbers. I.

For x = 2 : M n ( 2 ) = 4 P n − 1 + P n , where P n are Pell numbers.

II.

For x = 0 : M n ( 0 ) yields 4 , 1 , − 4 , − 1 , 4 , 1 , … , a period 4 sequence.

These special cases connect Mulatu polynomials to other well-studied sequences, facilitating comparative analysis.

An Efficient Algorithm for Detecting Mulatu Numbers

Building on perfect-square criteria for Fibonacci numbers, ⁹ we develop a similar criterion for Mulatu numbers.

Lemma:

If M is a Mulatu number, then 5 M 2 ± 76 is a perfect square.

The converse of the above lemma is not true.

Theorem:

N is a Mulatu number if and only if 5 N 2 ± 76 is a perfect square, and setting P = − N + 5 N 2 ± 76 , the iterative subtraction ( a , b ) → ( b , a − b ) starting from ( N , P ) eventually reaches ( 1 , 4 ) .

Proof:

From the Binet formula for Mulatu numbers ¹⁹: M n = ϕ n − ( − ϕ ) − n 5 + 3 F n − 1 , where ϕ = 1 + 5 2 .

Solving N = M n yields the quadratic Diophantine equation:

5 N 2 + 2 N + 1 = ( 2 F n + 1 ) 2 proving the necessity. Sufficiency follows by reconstructing n from F n = 5 N 2 + 2 N + 1 − 1 2 and verifying N = M n .

Theorem:

Polynomial Congruence

For all n ≥ 1 : M n ( x ) ≡ x n − 1 mod ( x 2 + x − 1 ) .

By induction using the recurrence M n ( x ) = x M n − 1 ( x ) + M n − 2 ( x ) and noting x 2 ≡ 1 − x mod ( x 2 + x − 1 ) . This result extends congruence properties known for Fibonacci polynomials ⁸ to the Mulatu case.

Theorem:

[GCD Property] For all n ≥ 1 , gcd ( M n , F n ) = 1 , where M n and F n are Mulatu and Fibonacci numbers respectively.

Proof:

From Lemma 1 in, ⁶ gcd ( M n , M n + 1 ) = 1 . Since F n = M n + 1 − M n , any common divisor of M n and F n divides M n + 1 , hence it must be 1.

This strengthens the relative primality results in ⁶ and parallels similar properties for Lucas numbers. ⁸

Theorem:

Odd-Indexed Sum. For n ≥ 1 , ∑ i = 1 n M 2 i − 1 ( x ) = M 2 n ( x ) − 4 . Telescoping sum using M 2 i − 1 ( x ) = M 2 i ( x ) − x M 2 i − 2 ( x ) , which follows from the recurrence.

This generalizes the summation formula for Mulatu numbers in ⁶ to polynomials.

The first seven Mulatu polynomials, computed using its recurrence relation, are as follows: n M n ( x ) M n ( 1 ) 0 4 4 1 1 1 2 x + 4 5 3 x 2 + x + 1 6 4 x 3 + 2 x 2 + 2 x + 4 11 5 x 4 + 3 x 3 + 4 x 2 + 3 x + 1 17 6 x 5 + 4 x 4 + 7 x 3 + 7 x 2 + 4 x + 4 28

These polynomials were verified using both the polynomial recurrence on Mulatu numbers and similar ideas on Fibonacci polynomials.

Power Congruences: Prime power congruence for Mulatu polynomials is established, analogous to known results for Fibonacci and Lucas polynomials. Theorem:

[Power Congruence] For prime p and n ≥ 0 :

M pn ( x ) = M n ( x p ) mod p , where M k ( x ) is the k th Mulatu polynomial defined by: M 0 ( x ) = 4 , M 1 ( x ) = 1 , M k ( x ) = x M k − 1 ( x ) + M k − 2 ( x ) . ( k ≥ 2 )

We also have: M n ( x ) = 2 L n ( x ) − F n ( x ) , where L n ( x ) and F n ( x ) are Lucas and Fibonacci polynomials respectively.

For Fibonacci polynomials: F pn ( x ) ≡ F n ( x p ) mod p

For Lucas polynomials: L pn ( x ) ≡ L n ( x p ) mod p

These are classical results (analogous to the integer case F pn ≡ F n mod p for Fibonacci numbers). The proof uses the binomial theorem and the fact that ( k p ) ≡ 0 mod ( p ) for 1 ≤ k ≤ p − 1 .

From M n ( x ) = 2 L n ( x ) − F n ( x ) , we have: M pn ( x ) = 2 L pn ( x ) − F pn ( x )

Apply the congruences: 2 L pn ( x ) ≡ 2 L n ( x p ) mod p and F pn ( x ) ≡ F n ( x p ) mod p

Thus: M pn ( x ) ≡ 2 L n ( x p ) − F n ( x p ) mod p M pn ( x ) ≡ M n ( x p ) mod p

3. Conclusion and future work

We have extended the theory of Mulatu numbers to the polynomial domain, introducing Mulatu polynomials and establishing their fundamental properties. Our work connects to and extends recent research on generalized Fibonacci sequences ^{1–
3} and polynomial recurrences. ⁴

Future research directions include: •

Investigating combinatorial interpretations of Mulatu polynomial coefficients, possibly connection to trails or paths as done for Fibonacci polynomials. ¹⁵

•

Generalizing to higher-order linear recurrences, following recent work on k-Fibonacci sequences. ¹²

•

Exploring cryptographic applications, building on uses of Fibonacci-like sequences in coding theory. ²³

•

Extending the detection algorithm to other linear recurrence sequences with arbitrary initial conditions.

•

Studying the distribution of Mulatu numbers in arithmetic progressions, following recent work on Fibonacci numbers. ²²

Declarations Ethics

We hereby declare that the information provided above is accurate, and that this research was conducted in compliance with all applicable ethical guidelines and institutional policies. Since this study did not involve any human or animal participants, there was no need for ethical approval or consent.

Originality

We declare that the research presented in this paper is our original work. This work has not been submitted for any other degree or qualification, and all the sources and references used have been appropriately acknowledged.

Data availability

Since it is a study on a mathematical theory, there is no extended data used.

References 1

Falcon

Plaza

: On the Fibonacci k-numbers. Chaos, Solitons Fractals. 2007;32(5):1615–1624. 10.1016/j.chaos.2006.09.022

Akkuş

Kuloğlu

Özkan

: Analytical Characterization of Self-Similarity in k-Cullen Sequences Through Generating Functions and Fibonacci Scaling. Fractal and Fractional. 2025;9(6). 10.3390/fractalfract9060380

Özkan

Akkuş

: Copper Ratio Obtained by Generalizing the Fibonacci Sequence. AIP Adv. 2024;14(7).

Webster

Logan

: Polynomial Generalizations of Fibonacci and Lucas Sequences. J Integer Seq. 2023;26.

Lemma

Lambright

Andreou

: The Mathematical Magic of the Mulatu Numbers. Arts & Humanities Hawaii. 2012;1–11.

Derso

Admasu

: More on the Fascinating Characterizations of Mulatu's Numbers. F1000Res. 2025;13:1306. 10.12688/f1000research.157738.2

Hoggatt

: Fibonacci and Lucas Numbers. Houghton Mifflin;1973.

Koshy

: Fibonacci and Lucas Numbers with Applications. Wiley-Interscience;2001.

Renault

: The Period, Rank, and Order of the (a,b)-Fibonacci Sequence Mod m. Math. Mag. 2013;86(5):372–380. 10.4169/math.mag.86.5.372

Meher

Swamy

MNS

: Efficient Algorithms for Testing Membership in Fibonacci and Lucas Sequences. IEEE Trans. Comput. 2020;69(5).

Adédji

Bachabi

Togbé

: On Thabit and Williams Numbers Base b as Sum or Difference of Two Repdigits. Math Pannon. 2024;30(2):142–161.

Marín

Jiménez

: Generalized Fibonacci Sequences and Their Matrix Representations. Linear Algebra Appl. 2022;647.

Bernstein

: Fast Multiplication and Its Applications. Algorithmic Number Theory. 2008;44:325–384. 10.1017/9781139049801.011

Horadam

: Basic Properties of a Certain Generalized Sequence of Numbers. Fibonacci Q. 1965;3:161–176. 10.1080/00150517.1965.12431416

Benjamin

Quinn

: Proofs That Really Count: The Art of Combinatorial Proof. Mathematical Association of America;2003.

Lemma

Lambright

Andreou

: The Mathematical Magic the Mulatu Numbers. Arts humanities hawaii. 2012;1–11.

Lemma

Lambright

: The fascinating mathematical identities relating the Mulatu and Fibonacci numbers. Adv. Appl. Math. Sci. 2012;11(7):285–290.

Galvez

Dehesa

: Novel Properties of Fibonacci and Lucas Polynomials. Math. Proc. Camb. Philos. Soc. 1985;97:159–164. 10.1017/S0305004100062708

Filipponi

Horadam

: Second Derivative Sequences of Fibonacci and Lucas Polynomials. Fibonacci Q. 1993;31(3):194–204. 10.1080/00150517.1993.12429280

Costa

Costa Mesquita

ÉG

da Catarino

PMMC

: On the connections between Fibonacci and Mulatu Numbers. Intermaths. 2025;6(1):17–36. 10.22481/intermaths.v6i1.16742

Stanley

: Enumerative Combinatorics. Cambridge University Press; 2nd ed 2011;1.

Lenstra

Shallit

: Continued Fractions and Linear Recurrence Sequences. J Theor Nombres Bordeaux. 2022;34(2).

Hoo

Madanshekaf

: Applications of Fibonacci Sequences in Coding Theory. J. Discret. Math. Sci. Cryptogr. 2021;24(2):457–470.

10.5256/f1000research.200418.r485762

Reviewer response for version 1

Batte

Herbert

1 Referee 1Makerere University, Kampala, Uganda

Competing interests: No competing interests were disclosed.

13 6 2026

2026

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

reject

Overall Recommendation: Major Revision.

The paper introduces Mulatu polynomials M_n(x) defined by M_n(x) = x*M_{n-1}(x) + M_{n-2}(x) with M_0(x) = 4, M_1(x) = 1, and claims to develop an O(log N) detection algorithm for Mulatu numbers. The polynomial generalization is a natural and legitimate direction; the table of the first seven Mulatu polynomials is correct, and the identity M_n(x) = 4F_{n-1}(x) + F_n(x) is valid and well proved. However, the paper contains several critical mathematical errors that prevent acceptance.

Major Issues

1. Self-contradictory detection theorem. On page 5 the authors state a Lemma asserting that the converse of the criterion “5M² ± 76 is a perfect square” is not true, then on page 6 a Theorem states the same criterion as an if-and-only-if characterisation. These two statements are in direct and irreconcilable contradiction. Since the detection algorithm — advertised in the title and abstract — rests on the Theorem, the paper’s main claimed contribution is undermined by the authors’ own preceding lemma. The authors must decide which statement is correct, supply a counterexample if the Lemma is retained, and provide a rigorous proof of whichever criterion is kept.

2. Invalid proof of the detection theorem. The proof on page 6 asserts, without derivation, that “solving N = M_n yields 5N² + 2N + 1 = (2F_n + 1)²”. No such derivation is given, the Binet formula does not yield this equation by standard manipulation, and the criterion in the proof (5N² + 2N + 1 a perfect square) is algebraically different from the criterion in the Theorem statement (5N² − 76 a perfect square). A complete derivation is required.

3. Incorrect GCD proof. The proof of gcd(M_n, F_n) = 1 (page 6) relies on the claim F_n = M_{n+1} − M_n. This is false: the Mulatu recurrence gives M_{n+1} − M_n = M_{n-1}, not F_n. The argument is also logically circular, invoking gcd(M_n, M_{n+1}) = 1 without stating it. The proof must be completely rewritten.

4. Incorrect period of M_n(0). The paper claims the sequence 4, 1, 4, 1, … has period 4. It has period 2. This must be corrected.

5. Incomplete proofs. The Corollary proof (M_n(x) = L_n(x) + 2F_{n-1}(x)) invokes the identity L_n(x) = F_{n-1}(x) + F_{n+1}(x) without proof or precise citation, and omits the algebraic steps. The polynomial congruence theorem (page 6) lacks base cases. The power congruence theorem (page 7) gives no reference for the Lucas/Fibonacci polynomial congruences, omits the binomial theorem argument, and does not discuss the required linearity mod p; additionally, the identity M_n(x) = 2L_n(x) − F_n(x) used critically in that proof is itself never proved or cited. No complexity analysis is provided for the claimed O(log N) algorithm.

Minor Issues

Results carry no systematic numbering, making cross-referencing impossible. Reference [9] (Renault 2013) is incorrect for the Fibonacci perfect-square criterion, which is due to earlier authors. Reference [11] (Adedji et al. 2024) concerns Thabit and Williams numbers and has no connection to Mulatu numbers. Reference [19] (Filipponi–Horadam) is misattributed for the Binet formula; the correct reference appears to be [20] (Costa et al., Intermaths 2025). The generating function is stated without derivation. The abstract mentions modular arithmetic but the body of the paper uses no modular arithmetic in the detection algorithm proof. Several grammatical issues are present (e.g. “combinatory” should be “combinatorics”).

The authors are encouraged to resubmit after: (1) resolving the contradiction between the Lemma and the detection Theorem and providing a correct, complete proof; (2) correcting the GCD proof; (3) completing all incomplete proofs and verifying all special cases; and (4) correcting citations and introducing systematic theorem numbering.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Are the conclusions drawn adequately supported by the results?

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Diophantine equations; linear recurrence sequences; Fibonacci, Lucas, and generalised linear recurrence polynomial sequences; analytic and computational number theory.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

10.5256/f1000research.200418.r488752

Reviewer response for version 1

Mesquita

Élis Gardel da Costa

1 Referee https://orcid.org/0000-0003-2385-4108 1Universidade Federal do Tocantins, Palmas, Brazil

Competing interests: No competing interests were disclosed.

13 6 2026

2026

recommendation

reject

Referee Report

The manuscript entitled {``Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers''} studies a polynomial extension of the classical Mulatu sequence and proposes an algorithm for detecting Mulatu numbers through a perfect-square criterion.

The topic is potentially interesting. Since Mulatu numbers satisfy the same recurrence as Fibonacci and Lucas numbers but possess distinct initial conditions, the study of their polynomial analogues is a natural direction. The manuscript also attempts to establish connections with Fibonacci polynomials, Lucas polynomials, generating functions, Binet-type formulas and arithmetic properties.

However, after a careful examination of the paper, I found several substantial mathematical issues affecting some of the main results. In particular, there are inconsistencies in definitions, incorrect identities, invalid proofs, and some statements that fail under direct verification. Consequently, the current version of the manuscript is not yet suitable for indexing.

My comments are detailed below.

Major Comments

1. Inconsistency in the definition of Lucas polynomials

In the literature review, the manuscript states that Lucas polynomials satisfy

L_0(x)=2,\qquad L_1(x)=1.

Later, in the preliminary section, the manuscript uses

L_0(x)=2,\qquad L_1(x)=x,

which is the standard definition.

This inconsistency must be corrected, since several subsequent identities depend on the adopted initial conditions.

2. Conceptual justification of Mulatu polynomials

The authors define

M_0(x)=4,\qquad M_1(x)=1,

and

M_n(x)=xM_{n-1}(x)+M_{n-2}(x).

This definition is mathematically legitimate and indeed satisfies

M_n(1)=M_n.

Nevertheless, the manuscript repeatedly states that this construction follows the pattern of Fibonacci and Lucas polynomials. In my opinion, this point deserves a more careful discussion.

For Fibonacci polynomials,

F_0(x)=0,\qquad F_1(x)=1,

while for Lucas polynomials,

L_0(x)=2,\qquad L_1(x)=x.

Therefore, an equally natural analogue for Mulatu numbers would be

M_0(x)=4,\qquad M_1(x)=x.

The authors should explain why the choice \(M_1(x)=1\) is preferred and perhaps discuss alternative polynomial versions. This would significantly strengthen the conceptual foundation of the paper.

3. Relation with Fibonacci polynomials

The identity

M_n(x)=4F_{n-1}(x)+F_n(x)

is correct and constitutes one of the strongest results in the manuscript.

However, this identity also reveals that the proposed Mulatu polynomials belong to the vector space generated by Fibonacci polynomials.

Therefore, the authors should better explain which genuinely new properties arise from the Mulatu setting and which results are direct consequences of known Fibonacci polynomial identities.

4. Incorrect identity involving Lucas polynomials

The manuscript claims

M_n(x)=L_n(x)+2F_{n-1}(x).

This identity is false.

Indeed,

L_n(x)=F_{n+1}(x)+F_{n-1}(x),

implies

L_n(x)+2F_{n-1}(x)

F_{n+1}(x)+3F_{n-1}(x)

xF_n(x)+4F_{n-1}(x).

On the other hand,

M_n(x)=4F_{n-1}(x)+F_n(x).

These expressions coincide only when \(x=1\).

Therefore the stated polynomial identity is incorrect and must be replaced by a valid relationship.

5. Detection theorem

The most serious issue in the manuscript concerns the proposed detection criterion.

The proof is based on the equation

5N^2+2N+1=(2F_n+1)^2.

This identity is false.

For example, taking \(N=M_1=1\),

5N^2+2N+1=8,

while

(2F_1+1)^2=9.

Thus the proof fails immediately.

Since the necessity part is incorrect, the sufficiency argument based on reconstructing \(F_n\) from

F_n=

\frac{\sqrt{5N^2+2N+1}-1}{2}

also collapses.

The entire detection theorem must therefore be reconsidered.

6. Iterative subtraction algorithm

The manuscript additionally claims that, after defining a quantity (P), the iteration

(a,b)\mapsto(b,a-b)

starting from \((N,P)\) eventually reaches \((1,4)\).

No convincing proof is provided.

Moreover, direct testing suggests that the statement cannot be correct as written.

This section requires substantial revision and a complete mathematical justification.

7. Polynomial congruence theorem

The manuscript states that

M_n(x)\equiv x^{n-1}\pmod{x^2+x-1}.

This statement is false.

Indeed,

M_2(x)=x+4,

while the claimed congruence would imply

M_2(x)\equiv x.

Since

M_2(x)-x=4

is not divisible by

x^2+x-1,

the theorem fails already for \(n=2\).

Furthermore, the induction argument presented in the manuscript does not establish the claimed result.

Therefore both the theorem and its proof must be revised.

8. GCD theorem

The proof of the GCD theorem relies on the implicit identity

F_n=M_{n+1}-M_n.

This identity is false.

For example,

M_4-M_3=11-6=5,

whereas

F_3=2.

Consequently, the current proof is invalid and must be replaced.

9. Odd-indexed summation formula

The manuscript states

\sum_{i=1}^{n}M_{2i-1}(x)=M_{2n}(x)-4.

Taking \(n=1\),

M_1(x)=1,

while

M_2(x)-4=x.

Hence the identity is false as a polynomial identity.

It only holds after specialization at \(x=1\).

This result should therefore be corrected.

\subsection*{10. Table of the first polynomials}

The list of the first Mulatu polynomials is inconsistent with the recurrence.

Using

M_0(x)=4,\qquad M_1(x)=1,

the recurrence yields

M_2(x)=x+4,

M_3(x)=x^2+4x+1,

M_4(x)=x^3+4x^2+2x+4.

The table in the manuscript gives different expressions.

This section must be recomputed.

Minor Comments

1. Replace all occurrences of \(F_n\) by \(F_n(x)\) whenever polynomial sequences are intended.

2. The notation for Lucas polynomials should be standardized throughout the manuscript.

3. Several displayed formulas should be placed in proper mathematical environments and aligned for readability.

4. Some statements appear without proof or with only a brief indication. Additional details are necessary.

5. The English presentation could be improved in several places, especially in theorem statements and transitions between sections.

6. Singular/plural usage should be revised (e.g., `Lucas polynomial'' versus `Lucas polynomials'').

Recommendation

The paper addresses a potentially interesting topic and contains ideas that may lead to worthwhile results. However, several of the principal theorems are currently incorrect or insufficiently justified.

For this reason, I cannot recommend indexing in its present form.

I recommend substantial revision, requiring a thorough mathematical reconsideration of the manuscript, particularly the sections concerning the detection criterion, polynomial identities, congruence results, and arithmetic properties. The work may be reconsidered after the correctness of the main results has been rigorously established.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Are the conclusions drawn adequately supported by the results?

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

My expertise includes:Number Theory;Linear Recurrence Sequences;Fibonacci, Lucas, Horadam and related sequences;Polynomial sequences associated with linear recurrences;Discrete Mathematics;Algebraic and combinatorial properties of recursive sequences.My assessment primarily focuses on the mathematical correctness of the definitions, identities, recurrence relations, polynomial constructions, proofs, and algorithmic claims presented in the manuscript. I did not evaluate the work from the perspective of computational complexity theory, software implementation, or applied cryptographic applications beyond the mathematical arguments provided.