Pith. sign in

REVIEW 2 minor 10 references

Valuation Separation for Coprime Lucas Products

T0 review · 0 major / 2 minor · reviewed 2026-06-29 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The number-field abc conjecture implies only finitely many Lucas terms have squarefree part supported on any fixed finite set of primes.

desk verdict The paper cleanly separates valuations in coprime Lucas products via strong divisibility and gets an abc-conditional finiteness result on squarefree parts, which is a precise but limited step in the Diophantine program. read the letter →

arxiv 2605.24909 v1 pith:QPDTWC7A submitted 2026-05-24 math.NT

Dongyeon Kym This is my paper · ORCID
classification math.NT
keywords LucassequencesDiophantineequationsabcconjecturesquarefreepartscoprimeindicesvaluationseparationstrongdivisibility
topics The abc Conjecture
open problems The abc 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 considers Diophantine equations in which a fixed integer A times a k-th power equals the product of terms from a nondegenerate Lucas sequence U_n(P, Q) with Q = ±1, where the indices are pairwise coprime. The strong divisibility property makes these terms pairwise coprime, so the global power condition on the product factors into independent valuation conditions on each U_{n_i}. For squares this yields a termwise restriction: each factor's signed squarefree part must be supported on the primes dividing A. Assuming the abc conjecture in the quadratic field Q(sqrt(Δ)), the paper proves there are only finitely many such Lucas terms for any fixed finite set of rational primes. This turns the original equations into an abc-conditional finite search over a finite list of candidates.

What carries the argument

The strong divisibility property gcd(U_m, U_n) = U_gcd(m,n), which forces the factors U_{n_i} to be pairwise coprime when the indices are pairwise coprime and thereby separates the global k-th power condition into independent local valuation conditions on each factor.

What would settle it

An explicit infinite family of indices n for which the squarefree part of U_n(P,Q) is supported only on a predetermined finite set of primes would show the main finiteness statement is false.

Watch

Extended reading notes

Core claim

Assuming the number-field abc conjecture over Q(√Δ), only finitely many terms U_n in a nondegenerate Lucas sequence with Q = ±1 and positive discriminant Δ have squarefree part supported on a fixed finite set of rational primes. Consequently the equations A y^k = product of U_{n_i} with pairwise coprime indices admit an abc-conditional finite reduction. The paper also records the corresponding statement for general k and notes a primitive-divisor obstruction.

Load-bearing premise

The number-field abc conjecture holds over the quadratic field Q(sqrt(Δ)).

Editorial extensions

If this is right

  • The equation Δ y^{2} = U_m U_n with gcd(m,n)=1 reduces to checking a finite list of square-class compatibilities plus an integrality condition.
  • The k-th power version of the coprime-product equation likewise reduces to finitely many cases under the same abc assumption.
  • A primitive-divisor obstruction further restricts possible solutions in these equations.

Reading between the lines

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

  • The same valuation-separation technique could be applied to other divisibility sequences that satisfy an analogous gcd identity.
  • An effective form of the abc conjecture would turn the finite reduction into an explicit algorithm for finding all solutions.
  • The result connects the distribution of squarefree parts in Lucas sequences to the abc conjecture in quadratic fields.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper shows that for nondegenerate Lucas sequences U_n(P,Q) with Q=±1 and Δ>0, the strong divisibility property implies that factors U_{n_i} with pairwise coprime indices are pairwise coprime. This allows global k-th power conditions on products ∏ U_{n_i} to separate into local valuation conditions on each term. For k=2 this yields termwise restrictions on signed squarefree parts. Assuming the number-field abc conjecture over Q(√Δ), the authors prove that only finitely many U_n have squarefree part supported on any fixed finite set of rational primes, yielding an abc-conditional finite reduction for the Diophantine equations; a k-th power analogue and primitive-divisor obstruction are also given.

Significance. If the number-field abc conjecture holds, the finiteness result supplies a concrete reduction for a family of Diophantine equations involving Lucas sequences, converting an a priori infinite search into a finite check once square-class compatibility is verified. The unconditional separation step rests only on the standard strong-divisibility property and is therefore immediately applicable. The work is a modest but clean contribution that isolates the precise point at which abc is needed.

minor comments (2)
  1. [§3] §3 (abc application): the precise formulation of the number-field abc conjecture used (including the dependence on the discriminant Δ) should be stated explicitly rather than referenced only by name, to make the reduction fully self-contained.
  2. The integrality condition mentioned after the square-class restriction for Δ y² = U_m U_n is not expanded; a brief sentence clarifying what this condition reduces to would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, assessment of significance, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; result is explicitly conditional on external conjecture

full rationale

The derivation begins from the unconditional strong divisibility property of nondegenerate Lucas sequences with Q=±1, which separates the global k-th power condition into local valuation conditions on coprime factors. The finiteness statement for squarefree parts supported on a fixed prime set is stated as conditional on the number-field abc conjecture over Q(√Δ), an external hypothesis not derived or fitted inside the paper. No self-definitional equations, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the provided structure. The argument therefore remains self-contained against external benchmarks and does not reduce to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The finiteness claim depends on the external abc conjecture; no free parameters or new entities are introduced in the abstract.

assumptions (1)
  • domain assumption The number-field abc conjecture over Q(√Δ)
    Invoked to prove that only finitely many Lucas terms have squarefree part supported on a fixed finite set of rational primes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Valuation Separation for Coprime Lucas Products." pith.science (2026). https://pith.science/paper/QPDTWC7A

@misc{pith2026260524909,
  author       = {Pith},
  title        = {Pith review of: Valuation Separation for Coprime Lucas Products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QPDTWC7A}},
  note         = {Machine review of arXiv:2605.24909}
}
abstract

Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\pm 1$ and discriminant $\Delta=P^2+4Q>0$. We study Diophantine equations \[ A y^k=\prod_{i=1}^r U_{n_i}(P,Q), \qquad k\geq 2, \] where the indices $n_1,\ldots,n_r$ are pairwise coprime. The strong divisibility property implies that the factors $U_{n_i}$ are pairwise coprime, and hence a global $k$-th power condition separates into local valuation conditions on the individual factors. For $k=2$, this gives a termwise square-class restriction: each $U_{n_i}$ has signed squarefree part supported on the primes dividing $A$. In particular, the equation $\Delta y^2=U_mU_n$, with $\gcd(m,n)=1$, reduces to a finite square-class compatibility condition together with an integrality condition. Assuming the number-field $abc$ conjecture over $\mathbb Q(\sqrt{\Delta})$, we prove that only finitely many Lucas terms have squarefree part supported on a fixed finite set of rational primes. Consequently, the coprime product equations above admit an $abc$-conditional finite reduction. We also give the corresponding $k$-th power analogue and a primitive-divisor obstruction.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [1]

    Y. Bilu, G. Hanrot and P. M. Voutier,Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math.539(2001), 75–122. DOI: 10.1515/crll.2001.080

    work page doi:10.1515/crll.2001.080 2001

  2. [2]

    Bugeaud, M

    Y. Bugeaud, M. Mignotte and S. Siksek,Classical and modular approaches to exponential Diophan- tine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2)163(2006), no. 3, 969–1018. DOI: 10.4007/annals.2006.163.969

    work page doi:10.4007/annals.2006.163.969 2006

  3. [3]

    R. D. Carmichael,On the numerical factors of the arithmetic formsα n ±β n, Ann. of Math. (2)15 (1913), no. 1–4, 30–70. DOI: 10.2307/1967797

    work page doi:10.2307/1967797 1913

  4. [4]

    J. H. E. Cohn,Square Fibonacci numbers, etc., Fibonacci Quart.2(1964), 109–113. DOI: 10.1080/00150517.1964.12431509

    work page doi:10.1080/00150517.1964.12431509 1964

  5. [5]

    Everest, A

    G. Everest, A. van der Poorten, I. Shparlinski and T. Ward,Recurrence Sequences, Math. Surveys Monogr.104, Amer. Math. Soc., Providence, RI, 2003. DOI: 10.1090/surv/104

    work page doi:10.1090/surv/104 2003

  6. [6]

    1983 , PAGES =

    S. Lang,Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983. DOI: 10.1007/978-1-4757-1810-2

    work page doi:10.1007/978-1-4757-1810-2 1983

  7. [7]

    Luca and T

    F. Luca and T. N. Shorey,Products of members of Lucas sequences with indices in an interval being a power, J. Number Theory129(2009), no. 2, 303–315. DOI: 10.1016/j.jnt.2008.06.015. 18 DONGYEON KYM

    work page doi:10.1016/j.jnt.2008.06.015 2009

  8. [8]

    Peth˝ o,Perfect powers in second order linear recurrences, J

    A. Peth˝ o,Perfect powers in second order linear recurrences, J. Number Theory15(1982), no. 1, 5–13. DOI: 10.1016/0022-314X(82)90079-8

    work page doi:10.1016/0022-314x(82)90079-8 1982

Show all 10 references
  1. [9]

    J. H. Silverman,Wieferich’s criterion and theabc-conjecture, J. Number Theory30(1988), no. 2, 226–237. DOI: 10.1016/0022-314X(88)90019-4

    1988 doi

  2. [10]

    Zsigmondy,Zur Theorie der Potenzreste, Monatsh

    K. Zsigmondy,Zur Theorie der Potenzreste, Monatsh. Math. Phys.3(1892), 265–284. DOI: 10.1007/BF01692444. Email address:kynteger@dgu.ac.kr

    doi

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.