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.