bibkey: matijasevick2026squares
authors: Mateo Matijasevick and Santiago Rodriguez and Gregorio Salazar
year: 2026
title: Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields
doi: 10.48550/arXiv.2610.00765
url: https://arxiv.org/abs/2610.00765v1
claim: Section 7 question 2 asks for a natural rational-recursion family with reduced numerator plus denominator a cube, outside the conjugacy class of the cubic power map.
strata_touched:
  - D5/S3/Arith/CubeRecursionFamily
  - D5/S3/Arith/CubeRecursionProjective
license: citation-only
triage: anchor

Cubes beyond the reciprocal-sum square families

The cited version’s Section 7, second closing question, reads:

For the recursions of this paper the exponent m=2 is forced. Is there a natural family of rational recursions for which c a_n+b_n is always a cube, other than those conjugate to z -> z^3?

Problem 1.1 uses coprime positive numerator and denominator, positive integer initial values, a positive rational constant c, and indices n at least 2. The question does not define “natural.” The construction documented in Problems/arxiv-2610-00765-cube-recursion-family.md uses the explicit minimal interpretation of a uniform autonomous rational family over Q with a nonnegative integer parameter, fixed degree, one formula, and one constant independent of parameter, initial value and index. An unstated stricter interpretation remains unverified.

The source’s square results concern its reciprocal-sum class and do not assert an impossibility for arbitrary rational maps. The family F_t in the repository is a repository-derived answer under the stated interpretation, not a theorem attributed to this paper. The supplied ordinary comparison inspected the paper’s source and compared the relevant conjugacy definitions in Jones–Manes, Galois theory of quadratic rational functions, arXiv:1101.4339v3, and the scope of Cahn–Jones–Spear, Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture, arXiv:1512.03085v2. Quadratic-map Galois questions and rational values that are powers do not supply the whole reduced-coordinate cube assertion.

No equivalent whole construction was identified in those checked statements. The source’s two companion preprints were not located by the supplied bounded search. Global priority is UNDETERMINED; no exhaustive novelty claim follows. An identified equivalent whole answer would require reuse and remove new-solve eligibility without changing the mathematical validity of the construction.

Verified locator

• Versioned source: https://arxiv.org/abs/2610.00765v1
• DOI: https://doi.org/10.48550/arXiv.2610.00765