Extension of the Equation $\sum\limits_{j=1}^{k}jF_{j}^{p}=F_{n}^{q}$ to a Family of Lucas Sequences

Benjamin Earp-Lynch; Omar Kihel; Puntani Pongsumpun; Simon Earp-Lynch

Extension of the Equation $\sum\limits_{j=1}^{k}jF_{j}^{p}=F_{n}^{q}$ to a Family of Lucas Sequences

T0 review · 0 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read The weighted power-sum equation for Lucas sequences U_n(x, ±1) has no solutions when k ≥ 3 and max{p, q} ≤ 11, and only five solutions when k = 2.

desk verdict Clean, correctly executed extension of known power-sum Diophantine results from Fibonacci/Pell to the full family U_n(x,±1), with a new complete k=2 analysis. read the letter →

arxiv 2607.09105 v1 pith:YYXP5JCS submitted 2026-07-10 math.NT

Benjamin Earp-Lynch , Simon Earp-Lynch , Omar Kihel , Puntani Pongsumpun This is my paper · ORCID

classification math.NT MSC 11B3911D6111D45

keywords LucassequencesFibonaccinumbersPellDiophantineequationpowersumsBinetformulaprimitivedivisors

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 asks when a weighted sum of p-th powers of terms from a Lucas sequence of the first kind can itself be a pure q-th power of a later term. The sequences considered are those with the second parameter fixed at y = ±1, so they include the Fibonacci and Pell numbers as special cases and many others. For the two-term sum (k = 2) the authors list every positive-integer solution without restricting the exponents. For longer sums they prove there are no solutions at all once both exponents are at most 11 and the first parameter x is large enough relative to y. The result therefore settles, inside a uniform family, several earlier Diophantine questions that had been treated only for single sequences, and it shows that the phenomenon of “almost no solutions” is not special to Fibonacci or Pell numbers.

What carries the argument

Binet-form approximation of the Lucas terms together with the p-adic valuation formula for U_n and a short computer search that reduces the analytic bound k < 838 to a finite list of 29 candidate tuples, all of which are then checked by hand.

What would settle it

Exhibit any single sextuple (n, k, p, q, x, y) with k ≥ 3, max{p, q} ≤ 11, y = ±1 and x large enough that satisfies the sum equation, or exhibit an extra solution of the two-term equation beyond the five listed tuples.

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Extended reading notes

Core claim

Theorem 1.1 asserts two things. First, the only positive-integer solutions of 1 + 2x^p = U_n(x, y)^q with y = ±1 are the five explicit tuples (4, p, 1, 1, -1), (1 + 2^{p+1}, p, 1, 2, 1), (3, 2, 2, 2, 1), (3, 1, 1, 2, -1) and (5, 3, 1, 3, 1). Second, when k ≥ 3, max{p, q} ≤ 11 and x is at least 2 (respectively 3) according as y = -1 (respectively +1), the full equation ∑_{j=1}^k j U_j(x, y)^p = U_n(x, y)^q has no solutions whatever.

Load-bearing premise

The final computer search that discards the last 29 candidate sextuples after the analytic bound has already been reduced must be free of overflow or incomplete enumeration.

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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

This paper does exactly what the title claims: it takes the sum equation that Gueth–Luca–Szalay and Tchammou–Togbé solved for Fibonacci and Pell, and settles it for every Lucas sequence U_n(x,±1) when max{p,q}≤11 and k≥3, plus a complete solution list for the k=2 case with no exponent restriction.

What is new is the uniform treatment. The k=2 analysis (Proposition 3.1) is self-contained and uses Zsigmondy plus Carmichael/primitive-divisor theorems in a way that genuinely needs y=±1; the five listed solutions look exhaustive. For k≥3 the argument re-uses the familiar Binet-plus-norm-plus-computer strategy, but the authors carefully re-derive the necessary growth lemmas (2.2–2.5) and the p-adic valuation control (via Sanna) so that the same bound machinery works for every x. The final computer reduction to 29 candidates under k<838 is finite and stated clearly; the stress-test correctly finds no analytic gap.

Soft spots are minor and proportional. The bound on k is looser than the pure Fibonacci case because of the need for uniformity, and the SageMath enumeration is not machine-checked, so an undetected overflow could in principle leave an extra solution. The authors themselves explain why |y|>1 breaks the method (constant-term divisibility becomes exponential in k), so the restriction is honest rather than artificial. Citations are appropriate and non-circular.

This is for people who work on Diophantine equations involving linear recurrences or power sums of special sequences. It is solid enough that a serious editor should send it to referees; the result is modest in scope but rigorously established inside its stated limits. I would cite the k=2 list if I needed it, and I would bring the paper to a reading group if we were discussing these techniques.

Referee Report

0 major / 5 minor

Summary. The paper solves the Diophantine equation ∑_{j=1}^k j U_j(x,y)^p = U_n(x,y)^q for Lucas sequences of the first kind with y=±1, positive integers x,p,q,k,n and max{p,q}≤11. Theorem 1.1 completely classifies the solutions of the k=2 case 1+2x^p=U_n(x,y)^q (five families) and proves that no solutions exist for k≥3 under the stated size restrictions on x relative to y. The argument proceeds via Binet-type growth bounds (Lemmas 2.2–2.5), a norm argument in Q(√D) that yields an explicit upper bound k<838 after the polynomial-divisibility restriction of Lemma 4.4, and a finite SageMath enumeration that eliminates the remaining 29 candidate tuples.

Significance. The work cleanly unifies and extends the earlier Fibonacci (Soydan–Németh–Szalay, Gueth–Luca–Szalay) and Pell (Tchammou–Togbé) results to the full two-parameter family U_n(x,±1). The k=2 classification is unconditional on the exponents and relies on classical tools (Mihăilescu, Zsigmondy, Carmichael/Bilu–Hanrot–Voutier). The computer-assisted non-existence proof for k≥3 is fully rigorous once the analytic bound is accepted; the finite search is small (k≤838, x≤10) and therefore reproducible. The paper also supplies a transparent explanation why the same method fails for |y|>1, which is useful for future work.

minor comments (5)

The abstract sentence is missing the word “in” (“We solve the equation … positive integers”). A quick grammatical pass would catch several similar slips (e.g., “EQUA TION”, “F AMILY” in the title).
In the statement of Theorem 1.1 the phrase “with x≥2 when y=−1 and x≥3 when y=1” appears only in the second half; it would be clearer to list the precise range of x for each y at the beginning of the theorem.
Lemma 2.7 is stated only for y=−1 and odd p; a one-sentence remark that the even-p case is already settled by the earlier norm argument would help the reader.
The final computer search (end of §4) reports that 29 candidates remain and that none work, but does not list them. A short table or a link to a SageMath worksheet would make the verification completely transparent.
References [7] and [8] are cited for the Fibonacci case with max{p,q}≤10; the paper claims the extension to exponent 11 is routine, yet never records the new numerical bound obtained after changing the constants. A single sentence would close the gap.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: bounds from Binet/identities plus finite computer search; no self-definitional or load-bearing self-citation chain.

full rationale

The derivation of Theorem 1.1 is self-contained against external mathematical facts. For k=2, Proposition 3.1 uses Mihăilescu, Zsigmondy, Carmichael/primitive-divisor theorems and Lemma 2.1 identities, then a finite SageMath check for n≤25, x≤8; none of these steps define the conclusion into the premises. For k≥3, Proposition 4.1 expands the sum via Binet, Lemmas 4.2–4.3 bound remainders from Lemmas 2.2–2.5 (themselves elementary consequences of Binet under |y+1|≤x), a norm argument in Q(√D) yields the explicit inequality (4.4)/(4.6), Lemma 4.4 supplies the polynomial bound x≤(k+1)²/4+1, and SageMath exhausts the resulting finite set (k bound reduced to <838, then 29 candidates checked). Prior special-case papers [8,11,15] are cited only as motivation; their authors do not overlap with the present authors, and the present proofs do not import a uniqueness theorem from the same group. There is no fitted parameter renamed as prediction, no ansatz smuggled via self-citation, and no equation that equals its input by construction. The only non-formal step is a finite, fully stated computer enumeration, which is independent evidence rather than circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Pure existence/non-existence theorem in elementary number theory. Relies only on classical theorems about Lucas sequences and a finite computer search; no free parameters or invented entities are introduced.

assumptions (5)

standard math Binet formulae and the standard identities for Lucas sequences of the first and second kinds (Lemma 2.1)
Used throughout §§2–4 to obtain growth bounds and closed-form expressions.
standard math Sanna’s p-adic valuation formula for U_n (Lemma 2.6)
Invoked in the proof of Lemma 2.7 to control the exact power of the discriminant dividing U_p.
standard math Zsigmondy’s theorem and Carmichael’s primitive-divisor theorem
Applied in the k=2 case (Proposition 3.1) to force nพ25 and then to a short computer check.
standard math Mihăilescu’s theorem (Catalan’s conjecture)
Used once for the equation 1+2^{p+1}=U_n^q when x=2.
domain assumption Restriction y=±1 and max{p,q}พ11
Explicitly stated in Theorem 1.1; the methods fail for |y|>1 because the constant-term argument of Lemma 4.4 no longer yields a polynomial bound on x.

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Pith. "Pith review of Extension of the Equation $\sum\limits_{j=1}^{k}jF_{j}^{p}=F_{n}^{q}$ to a Family of Lucas Sequences." pith.science (2026). https://pith.science/paper/YYXP5JCS

@misc{pith2026260709105,
  author       = {Pith},
  title        = {Pith review of: Extension of the Equation $\sum\limits_j=1^kjF_j^p=F_n^q$ to a Family of Lucas Sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YYXP5JCS}},
  note         = {Machine review of arXiv:2607.09105}
}

abstract

We solve the equation $\sum\limits_{j=1}^{k}jU_{j}(x,y)^{p}=U_{n}(x,y)^{q}$ positive integers $x,p,q,k,n$, with $y=\pm1$ and $\max\{p,q\}\leq11$, where $U_{m}(x,y)=\frac{\alpha^{m}-\beta^{m}}{\alpha-\beta}$ for $\alpha$ and $\beta$ roots of the polynomial $t^2-xt+y$. This generalizes existing results on similar equations, wherein the sequence was fixed as either the Fibonacci or Pell numbers. In addition, we find all solutions with $k=2$ and $y=\pm1$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 2 canonical work pages

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Yu. Bilu, G. Hanrot and P.M. Voutier,Existence of primitive divisors of Lucas and Lehmer numbers,J. Reine Angew. Math.539, 2001, 75–122

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E. Tchammou, A. Togb´ e,On the Diophantine equation Pk j=1 jP p j =P q n. Acta Math. Hungar. (2020). Department of Mathematics, Carleton University, K1S 5B6, Canada Email address:benjaminearplynch@cmail.carleton.ca Email address:simonearplynch@cmail.carleton.ca Department of M...

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Pith tools

Reviewed July 13, 2026 · model on record in the stance chip above.

[1] [1]

M. A. Bennett, K. Gy˝ ory, and ´A. Pint´ er,On the Diophantine equation1 k + 2k +· · ·+x k =y n, Compos. Math. 140 (2004), 1417–1431

2004

[2] [2]

Billal and S

M. Billal and S. Riasat,Integer Sequences. Divisibility, Lucas and Lehmer Sequences,Springer 2021

2021

[3] [3]

Yu. Bilu, G. Hanrot and P.M. Voutier,Existence of primitive divisors of Lucas and Lehmer numbers,J. Reine Angew. Math.539, 2001, 75–122

2001

[4] [4]

Carmichael,On the numerical factors of the arithmetic formsα n ±β n,Ann

P.D. Carmichael,On the numerical factors of the arithmetic formsα n ±β n,Ann. Math. (2)5(1904), 173-180. EXTENSION OF THE EQUATION kP j=1 jF p j =F q n TO A F AMILY OF LUCAS SEQUENCES 17

1904

[5] [5]

Gy˝ ory and´A

K. Gy˝ ory and´A. Pint´ er,On the equation1k + 2k +· · ·+x k =y n, Publ. Math. Debrecen62(2003), no. 3-4, 403–414, DOI 10.5486/pmd.2003.2919. Dedicated to Professor Lajos Tam´ assy on the occasion of his 80th birthday. MR2008104

work page doi:10.5486/pmd.2003.2919 2003

[6] [6]

Hajdu,On a conjecture of Sch¨ affer concerning the equation1 k +· · ·+x k =y n, J

L. Hajdu,On a conjecture of Sch¨ affer concerning the equation1 k +· · ·+x k =y n, J. Number Theory155 (2015), 129–138, DOI 10.1016/j.jnt.2015.03.015. MR3349441

work page doi:10.1016/j.jnt.2015.03.015 2015

[7] [7]

Altassan, F

A. Altassan, F. Luca,On the equation Pk j=1 jF p j =F q n. J. Number Theory217(2020), 256-277

2020

[8] [8]

Gueth, F

K. Gueth, F. Luca and L. Szalay.On a Diophantine equation involving powers of Fibonacci numbers. Proc. Japan Acad. Ser. A Math. Sci.96(2020), no. 4, 33-37

2020