On a Diophantine Equation Involving Lucas Numbers

Nazim I. Mahmudov; Seyran S. Ibrahimov

REVIEW 3 major objections 4 minor 13 references

On a Diophantine Equation Involving Lucas Numbers

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read No Lucas number is a sum of two powers of a single Lucas number.

desk verdict New result, clean n=1 case, but the n≥2 reduction has an unjustified inequality at (3.12) and missing numerical data; deserves a careful referee. read the letter →

arxiv 2506.14379 v4 pith:PDK2TWNX submitted 2025-06-17 math.NT

Seyran S. Ibrahimov , Nazim I. Mahmudov This is my paper · ORCID

classification math.NT MSC 11B3911D6111B83

keywords LucasnumbersDiophantineequationsexponentiallinearformsinlogarithmscontinuedfractionsexactdivisibilityprimitivedivisorscomputer-assistedproof

topics The Collatz Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish a complete nonexistence result: the exponential Diophantine equation $L_m^{n+k}+L_m^n = L_r$, where $L_t$ is the $t$-th Lucas number, has no positive-integer solutions $r,m,n,k$ with $m\ge 2$. If true, this settles the question of when a Lucas number can be written as a sum of two powers of the same Lucas number. The proof splits into two regimes with different tools: $n=1$ is handled by an exact factorization of differences of Lucas numbers together with the primitive divisor theorem, while $n\ge 2$ uses an exact divisibility lemma, a lower bound for linear forms in logarithms, a continued-fraction criterion, and a finite computer search. The result matters because it closes this whole family of equations rather than merely bounding the solutions.

What carries the argument

The engine of the proof is a chain of effective estimates. The starting point is the closed form $L_t=\alpha^t+\beta^t$ with $\alpha=(1+\sqrt5)/2$, which yields the inequalities $\alpha^{t-1}\le L_t < \alpha^{t+1}$. The $n\ge 2$ branch begins with the exact divisibility statement that $L_m^n\parallel L_r$ forces $L_m^{n-1}\parallel r/m$, giving the inequality $\log m + (n-1)(m-1)\log\alpha \le \log r$; this is what bounds $m$ and $n$. The equation itself is then rewritten as $0<\alpha^r L_m^{-(n+k)} - 1 < 1.0025/L_m^k$, a small nonzero linear form in logarithms, to which an explicit lower bound is applied, yielding crude upper bounds on $k$ and $r$. A continued-fraction criterion then sharpens the bound to $k\le 8$, and the remaining finite box is enumerated by computer.

What would settle it

Run the finite check the proof reduces to -- $2\le m\le 12$, $2\le n\le 16$, $1\le k\le 8$, which force $r\le 313$ -- looking for $(m,n,k,r)$ with $L_m^{n+k}+L_m^n=L_r$; one hit refutes the theorem, while confirming the full search closes the proof.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1: for every $m\ge 2$, no positive integers $n,k,r$ satisfy $L_m^{n+k}+L_m^n=L_r$. In the $n=1$ case the proof shows that $L_m$ must divide the difference $L_r-L_m$ and uses a factorization depending on $r-m \bmod 4$ to force either a factor of $5$ dividing a Lucas number (impossible) or a primitive prime divisor of a large Lucas factor dividing a smaller one (also impossible), leaving only two finite pairs to check. For $n\ge 2$, the proof uses an exact divisibility property to bound $m$ and $n$, then an explicit lower bound for linear forms in logarithms to bound $k$, then a continued-fraction criterion to push $k$ down to at most $8$, and finally an explicit computer search. The theorem is presented as complete: after the reductions only a finite range remains, and the authors report checking it exhaustively.

Load-bearing premise

The whole $n\ge 2$ argument rests on an imported lemma, unproved in this paper, saying that if $L_m^n$ exactly divides $L_r$, then $L_m^{n-1}$ exactly divides $r/m$.

Editorial extensions

If this is right

The equation $L_m^{n+k}+L_m^n=L_r$ is completely resolved: no solution exists for any base $m\ge 2$, so no search for large exotic solutions is needed.
In the $n=1$ subcase, the only candidate pairs surviving the divisibility analysis are $(m,r)=(2,10)$ and $(3,9)$, and the latter fails by direct evaluation.
When $L_m$ happens to be prime, the theorem gives a complete existence answer for a special case of equations of the form $w_t=p^a+p^b$ with $p$ prime.
The proof yields explicit effective bounds: after reduction, any hypothetical solution must satisfy $m\le 12$, $n\le 16$, $k\le 8$, and $r\le 313$, making the claim checkable by a finite computation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension implicit in the method is that the analogous Fibonacci equation $F_m^{n+k}+F_m^n=F_r$ should be approachable by the same route, since primitive divisors, exact divisibility, and continued-fraction reduction all have Fibonacci counterparts.
The only externally imported ingredient in the $n\ge 2$ branch is the exact divisibility lemma; if a comparable lemma is proved for other Lucas sequences of the first and second kinds, the paper's open equations (4.1) and (4.2) would likely be solvable for fixed parameters $a,b$ by the same scheme.
The numerical constants in the linear-forms bound are very conservative, so the final search range could probably be shrunk well below $r\le 313$ by a sharper computation; the structural nonexistence result would be unchanged.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: the theorem is new, the n=1 case is genuinely clean, and the overall strategy is standard-but-competent. The n≥2 proof, however, has a real gap at inequality (3.12), and unless the authors supply the missing numerical data, the paper does not establish the claimed reduction.

What the paper does well: the n=1 argument using the factorization identity and Carmichael's primitive divisor theorem is neat and correct. The exceptional cases are handled honestly. The application of Lemma 2.4 is legitimate, since gcd(L_m, L_m^k+1)=1, so L_m^n || L_r follows from the equation. The authors also ship Python code on Zenodo, which is more than many papers in this area do.

The soft spot is exactly where the stress test points. Combining (3.10) and (3.11) gives L_m^k < 1.0025 (J^(m)(S)+2)(n+k)/log α. At that point in the proof the only available bound is n≤61, so the right side should be (61+k), not (28+k). The number 28 appears without justification. If this is a typo, the intended argument might still work, but as printed, (3.12) is stronger than what has been proved. The subsequent conclusion k≤8 also depends on unlisted J^(m)(S) values. The Zenodo code may contain them, but the paper does not show the output or describe how the N_m were selected, so a reader cannot independently verify the reduction to r≤3865. This is a load-bearing gap, not a cosmetic omission.

The rest of the argument—the Matveev bounds, the derivation of m≤55 and n≤61, and the final finite check—looks plausible. Remark 3.1 is odd but harmless. The earlier concern about Lemma 2.4 does not hold up; that lemma is applied correctly.

Who is this for? Specialists in Diophantine equations with Lucas and Fibonacci sequences. They will want to know this equation is settled, and the n=1 proof is worth reading. But the paper currently needs one substantive correction and one transparency fix. I would send it to a serious referee with instructions to check the (3.12) step carefully and to request the code output for the J values.

Referee Report

3 major / 4 minor

Summary. The paper claims to settle the Diophantine equation L_m^{n+k} + L_m^n = L_r completely: for positive integers r, m, n, k with m >= 2, the authors assert there are no solutions. The proof splits into the case n = 1, handled by an exact factorization of L_r - L_m together with Carmichael's primitive divisor theorem, and the case n >= 2, handled by an exact divisibility lemma, Matveev's lower bound for linear forms in logarithms, a Legendre/continued-fraction reduction, and a final finite computer check. The n = 1 argument is clean and self-contained. For n >= 2, the structure is standard, but the reduction step contains several unverified and, as written, incorrect ingredients.

Significance. If the proof were correct, the result would be a definitive non-existence theorem for a natural exponential Diophantine equation involving Lucas numbers, complementing recent results on Fibonacci and Lucas equations. The n = 1 part is a strong, complete contribution. The n >= 2 part uses standard and appropriate tools (Matveev, exact divisibility, Legendre) with no fitted parameters, and the authors provide a Zenodo link for the computational part. However, the current manuscript does not establish the key reduction to a finite search, because a central inequality is unjustified and the stated continued-fraction lemma is false in the form used.

major comments (3)

[Section 3, Eq. (3.12)] Inequality (3.12) is not justified. At that point in the proof the only established bound is n <= 61 from (3.8), so combining (3.10) and (3.11) yields L_m^k < 1.0025 (J^{(m)}(S)+2) (n+k)/log alpha, and hence at best L_m^k < 1.0025 (J^{(m)}(S)+2) (61+k)/log alpha. The printed inequality (3.12) contains (28+k), which is not derivable from any bound appearing before it. Since the subsequent conclusion k <= 8 depends precisely on this inequality, the reduction to the finite range r <= 3865 is not established as written. The authors must either prove a bound n <= 28 at that stage or redo the numerical step with (61+k) and report the resulting bounds.
[Lemma 2.5 and Eq. (3.11)] Lemma 2.5 as stated is false. The lower bound is asserted to hold with J(S) := max{a_i : i = 0, 1, ..., N}, where q_N > S, but the correct continued-fraction lower bound requires the next partial quotient a_{N+1}, not the maximum of the previous ones. For example, for x = pi = [3; 7, 15, 1, 292, ...], take N = 1, S = 10, q_1 = 7 > S, and p/q = 22/7. Then |pi - 22/7| ≈ 0.00126, which is less than 1/(2*7^2), yet the claimed bound with J(S) = max{3,7} = 7 would give |pi - 22/7| > 1/((7+2)*7^2) ≈ 0.00227, a contradiction. Thus inequality (3.11), which is essential for deriving (3.12) and ultimately k <= 8, rests on an invalid lemma. The authors must correct the statement of Lemma 2.5, recompute the partial-quotient maxima accordingly, and verify that the numerical conclusion k <= 8 still follows.
[Section 3, finite verification after (3.12)] The step "By taking m = 2, 3, ..., 55 in (3.12), and using the corresponding values of J^{(m)}(S), we find that k <= 8" is not verifiable from the manuscript. No values of N_m or J^{(m)}(S) are listed, and no code output or table is provided. This numerical step is load-bearing: it shrinks the search space from k < 1.71*10^11 to k <= 8, which then yields r <= 3865 and the final finite range. The authors should include a table of the relevant continued-fraction data (or the code output) so that the reduction can be checked independently, especially because the two preceding issues affect exactly this computation.

minor comments (4)

[Section 3, Matveev application] The displayed Matveev inequality contains a missing closing parenthesis: the exponent reads "(1 + log(2r) logLm" and should be "(1 + log(2r)) logLm" or similar; please correct the typo.
[Section 3, n = 1 case] The sentence "It is clear that L_r is even, and therefore 3|r" would benefit from a brief justification: since L_m^{k+1} + L_m is even, L_r is even, and Lucas numbers are even exactly when their index is divisible by 3.
[Remark 3.1] Remark 3.1 is confusing: the phrase "the upper bound for r in inequality (3.6) is rounded up, so the inequality may not hold for values of r near this bound" could be read as suggesting that a legitimate bound may fail; since all subsequent functions are increasing, the remark is unnecessary and should be removed or rewritten to state the simple monotonicity argument.
[General presentation] The notation in Lemma 2.5 is ambiguous because J(S) is defined in terms of N inside the same lemma; if the lemma is corrected, please define N as the index satisfying q_N > S and state clearly whether J(S) includes a_{N+1}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation relies on external theorems (Carmichael, Matveev, Legendre, exact divisibility from Onphaeng–Pongsriiam) and a finite computation; no fitted parameter or self-citation chain is load-bearing.

full rationale

The paper's proof chain is not circular. For n = 1, the argument uses the factorization identity for differences of Lucas numbers (Lemma 2.1) and Carmichael's Primitive Divisor Theorem (Lemma 2.3), both external and independent of the target result. For n ≥ 2, the proof invokes Matveev's lower bound for linear forms in logarithms, Legendre's lemma on continued fractions, and Lemma 2.4 on exact divisibility imported from Onphaeng and Pongsriiam [11]; that lemma is an external published result whose hypotheses are verified from equation (1.1), not an input disguised as a conclusion. The only self-citation is reference [2], the authors' earlier related work on a Fibonacci equation, and it is cited merely as related literature, not as a premise for Theorem 3.1. There are no fitted parameters, no quantity that is predicted from the same data used to define it, and no uniqueness claim imported from the authors' own prior work. The skeptical concern about inequality (3.12) and unlisted J^(m)(S) values is a possible correctness or numerical-reproducibility gap, not a circularity: an unjustified bound or missing code output does not show that the theorem is equivalent to its assumptions. Under the stated rubric, no circular step can be quoted, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard external theorems and one specialized exact-divisibility lemma from [11]. No free parameters are fitted and no new entities are postulated; the only computation is a finite search for m <= 12, n <= 16, k <= 8, r <= 313 plus continued fraction maxima, with code deposited on Zenodo.

assumptions (6)

standard math Binet formula L_t = alpha^t + beta^t and the inequality alpha^(t-1) <= L_t < alpha^(t+1) (Koshy [4], equations (1.2)-(1.3))
Used throughout to convert equalities into size inequalities for r, m, n, k. Standard for Lucas sequences.
domain assumption Divisibility criterion L_m | L_r iff r = m(2s-1) for some integer s >= 2 (Lemma 2.2, citing Koshy [4])
Used in the n = 1 case to write r = m(2s-1), making r-m even and enabling Lemma 2.1.
standard math Carmichael Primitive Divisor Theorem (Lemma 2.3)
Used in the n = 1 case to produce a prime divisor of L_((r+m)/2) that does not divide L_m.
domain assumption Exact divisibility lemma: if L_m^n || L_r then m is not congruent to 0 mod 3, r/m is odd, and L_m^(n-1) || r/m (Lemma 2.4, citing [11])
Load-bearing for n >= 2: it yields inequality (2.1) and the bounds on m and n. The paper does not prove this lemma.
standard math Legendre's continued fraction lemma (Lemma 2.5)
Used to convert the lower bound on |x_m - r/(n+k)| into the final size bound on L_m^k.
standard math Matveev's lower bound for linear forms in logarithms (Theorem 2.1)
Provides the initial finite upper bound for k in terms of r.

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Cite this review

Pith. "Pith review of On a Diophantine Equation Involving Lucas Numbers." pith.science (2026). https://pith.science/paper/PDK2TWNX

@misc{pith2026250614379,
  author       = {Pith},
  title        = {Pith review of: On a Diophantine Equation Involving Lucas Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDK2TWNX}},
  note         = {Machine review of arXiv:2506.14379}
}

read the original abstract

Let L_t denote the t-th Lucas number. We prove that the Diophantine equation L_m^{n+k} + L_m^n = L_r has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Carmichael, R.D.: On the numerical factors of the arithmetic formsαn ±β n. Ann. Math.15, 30–70 (1913)

work page 1913
[2]

On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$

Ibrahimov, S.S., Mahmudov, N.I.: On the Diophantine equationFn =F k l (F m l −1). arXiv:2409.02047 (2024)

work page Pith review arXiv 2024
[3]

Kebli, S., Kihel, O., Larone, J., Luca, F.: OnthenonnegativeintegersolutionstotheequationFn±Fm = ya. J. Number Theory220, 107–127 (2021)

work page 2021
[4]

Wiley, New York (2001)

Koshy, T.:Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001)

work page 2001
[5]

Fibonacci Q.52(4), 290–295 (2014)

Laishram, S., Luca, F.: Fibonacci numbers of the formxa ±x b ±1. Fibonacci Q.52(4), 290–295 (2014)

work page 2014
[6]

Duprat, Paris (1798)

Legendre, A.M.:Essai sur la théorie des nombres. Duprat, Paris (1798)

work page
[7]

Luca, F., Patel, V.: On perfect powers that are sums of two Fibonacci numbers. J. Number Theory 189, 90–96 (2018)

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Fibonacci Q.45, 98–103 (2007)

Luca, F., Szalay, L.: Fibonacci numbers of the formpa ±p b + 1. Fibonacci Q.45, 98–103 (2007)

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Show all 13 references

[9]

Luca, F., Stănică, P.: Fibonacci numbers of the formpa ±p b. Congr. Numer.194, 177–183 (2009)

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[10]

Matveev, E.M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II. Izv. Ross. Akad. Nauk Ser. Mat.64, 125–180 (2000); English transl.: Izv. Math. 64, 1217–1269 (2000)

2000
[11]

AIMS Math.6(11), 11733–11748 (2021)

Onphaeng, K., Pongsriiam, P.: Exact divisibility by powers of the integers in the Lucas sequences of the first and second kinds. AIMS Math.6(11), 11733–11748 (2021)

2021
[12]

Fibonacci Q.56(4), 296–302 (2018)

Onphaeng, K., Pongsriiam, P.: The converse of exact divisibility by powers of the Fibonacci and Lucas numbers. Fibonacci Q.56(4), 296–302 (2018)

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[13]

Panda, G.K., Patra, A.: Exact divisibility by powers of the Pell and associated Pell numbers. Proc. Math. Sci.131, 20 (2021) 8

2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Carmichael, R.D.: On the numerical factors of the arithmetic formsαn ±β n. Ann. Math.15, 30–70 (1913)

work page 1913

[2] [2]

On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$

Ibrahimov, S.S., Mahmudov, N.I.: On the Diophantine equationFn =F k l (F m l −1). arXiv:2409.02047 (2024)

work page Pith review arXiv 2024

[3] [3]

Kebli, S., Kihel, O., Larone, J., Luca, F.: OnthenonnegativeintegersolutionstotheequationFn±Fm = ya. J. Number Theory220, 107–127 (2021)

work page 2021

[4] [4]

Wiley, New York (2001)

Koshy, T.:Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001)

work page 2001

[5] [5]

Fibonacci Q.52(4), 290–295 (2014)

Laishram, S., Luca, F.: Fibonacci numbers of the formxa ±x b ±1. Fibonacci Q.52(4), 290–295 (2014)

work page 2014

[6] [6]

Duprat, Paris (1798)

Legendre, A.M.:Essai sur la théorie des nombres. Duprat, Paris (1798)

work page

[7] [7]

Luca, F., Patel, V.: On perfect powers that are sums of two Fibonacci numbers. J. Number Theory 189, 90–96 (2018)

work page 2018

[8] [8]

Fibonacci Q.45, 98–103 (2007)

Luca, F., Szalay, L.: Fibonacci numbers of the formpa ±p b + 1. Fibonacci Q.45, 98–103 (2007)

work page 2007