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The Frobenius number corresponding to the squares of three consecutive Fibonacci numbers: comparison of three algorithmic processes

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Pith's one-line read For every integer $n \ge 4$, the Frobenius number of the numerical semigroup generated by the squares of the three consecutive Fibonacci numbers $f_n, f_{n+1}, f_{n+2}$ is exactly one of three explicit closed forms, determined by which of…

desk verdict Correct new closed form for a natural Frobenius family; the main gap (the Apéry-set inclusion) is routine to fill, but the advertised three-algorithm comparison is largely conjectural as written. read the letter →

arxiv 2507.01898 v1 pith:ZO4VV4IN submitted 2025-07-02 math.NT math.GR

Aureliano M. Robles-Pérez , José Carlos Rosales This is my paper
classification math.NTmath.GR MSC 11D0720M1320M14
keywords FrobeniusnumbernumericalsemigroupFibonaccinumbersApérysetclosed-formformulathreeconsecutivesquarespseudo-Frobeniusproblem
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 addresses the classical Frobenius coin problem for one concrete infinite family: the numerical semigroup $S(n) = \langle f_n^2, f_{n+1}^2, f_{n+2}^2 \rangle$ generated by the squares of three consecutive Fibonacci numbers. Its goal is to determine the largest integer not representable as a nonnegative sum of those three squares, and it establishes that for every $n \ge 4$ this Frobenius number is given by one of three explicit closed formulas, selected by which of $f_n, f_{n+1}, f_{n+2}$ is even. The proof describes the Apéry set of $S(n)$ modulo $f_n^2$ exactly as a finite box of representative pairs with one corner removed, from which the pseudo-Frobenius numbers and then the Frobenius number follow directly. Three algorithmic routes are compared; the first two yield the same formulas with some steps left as claims, while the third route supplies the fully proved derivation. If the result is right, an entire infinite family of otherwise-hard Frobenius problems is resolved by a parameter-free formula.

What carries the argument

The central object is the Apéry set $\operatorname{Ap}(S(n), f_n^2)$, the least nonnegative representative in each residue class modulo $f_n^2$ inside $S(n)$; it carries the argument through the standard identity $F(S) = \max \operatorname{Ap}(S, f_n^2) - f_n^2$ and the description of pseudo-Frobenius numbers as maximal elements of the Apéry set under the semigroup order. In each parity case the paper proves this set is exactly $\{\lambda f_{n+1}^2 + \mu f_{n+2}^2 : (\lambda, \mu) \in C_1 \times C_2 \setminus C_3 \times C_4\}$, a product box with one corner removed, with the boxes specified in Propositions 5.7, 5.10, and 5.13. Three Fibonacci identities are refined by the parity of $f_n$, $f_{n+1}$, and $f_{n+2}$ to produce the three box shapes, and a cardinality count converts inclusion into equality. This box description is what turns a search problem into a closed-form computation.

What would settle it

For any fixed $n \ge 4$, enumerate $\lambda f_{n+1}^2 + \mu f_{n+2}^2$ over the claimed pair set, reduce modulo $f_n^2$, and check that each residue class occurs exactly once and that the largest value equals the closed form in Corollary 5.16; a single $n$ where either check fails would falsify the Apéry-set description and hence the formula.

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Extended reading notes

Core claim

The paper's central claim, proved as Corollary 5.16, is that for every $n \ge 4$ the Frobenius number of $S(n) = \langle f_n^2, f_{n+1}^2, f_{n+2}^2 \rangle$ is exactly one of three closed formulas. If $f_n$ is even, $F(S(n)) = \left(\frac{f_{n+3}}{2}-1\right) f_{n+1}^2 + (f_{n-2}-1) f_{n+2}^2 - f_n^2$. If $f_{n+1}$ is even, $F(S(n)) = (f_{n+2}-1) f_{n+1}^2 + \left(\frac{f_{n-2}}{2}-1\right) f_{n+2}^2 - f_n^2$. If $f_{n+2}$ is even, $F(S(n)) = \left(\frac{f_{n+2}}{2}-1\right) f_{n+1}^2 + \left(\frac{f_{n-2}+f_n}{2}-1\right) f_{n+2}^2 - f_n^2$. In each case the pseudo-Frobenius set consists of two elements, one of which is the Frobenius number. The derivation fixes the Apéry set modulo $f_n^2$ as a rectangle $C_1 \times C_2$ with a corner $C_3 \times C_4$ removed, giving exactly $f_n^2$ representatives, and the maximum of these representatives minus $f_n^2$ is the desired number. The three algorithmic processes of the paper all land on the same formulas, but only the Apéry-set route is developed with complete proofs.

Load-bearing premise

The load-bearing premise is that the Apéry set modulo $f_n^2$ consists exactly of the displayed rectangle-minus-corner pairs in each parity case, with the direction 'every minimal representative lies in the box' asserted but not fully derived in the text.

Editorial extensions

If this is right

  • For every $n \ge 4$, $F(S(n))$ can be evaluated directly from Fibonacci values, with no search or iterative algorithm.
  • The pseudo-Frobenius set of $S(n)$ has exactly two elements in each parity case, so every semigroup in the family has type $2$.
  • The formulas obtained from the first two algorithmic routes coincide with the proved formulas, so the unproved claims in Sections 3 and 4 are corroborated for this family.
  • The family provides a new infinite class with explicit Frobenius numbers, a useful testing ground in a problem known to have no uniform polynomial formula in three variables.

Reading between the lines

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

  • Editorial extension: the same rectangle-minus-corner Apéry description may hold for squares of consecutive terms of other two-term linear recurrences, so one could test a Lucas-square analogue of the three formulas.
  • Editorial extension: because all three routes agree, this family can serve as a concrete benchmark for locating precisely where the unproved steps of the first two algorithms would need added hypotheses.
  • Editorial extension: carrying the same Apéry-box method to cubes rather than squares is a direct next test; the parity splits would change, but the box shape may persist.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper computes the Frobenius number F(f_n^2, f_{n+1}^2, f_{n+2}^2) for n ≥ 4, where f_i is the Fibonacci sequence, by comparing three algorithmic processes: the Ramírez Alfonsín–Rødseth Apéry-set algorithm, Tripathi's three-variable formulas, and the Rosales–García-Sánchez Apéry-set method. In Section 5, the main result (Corollary 5.16) states that F(S(n)) is one of three explicit closed forms depending on whether f_n, f_{n+1}, or f_{n+2} is even, with S(n) = ⟨f_n^2, f_{n+1}^2, f_{n+2}^2⟩. Sections 3 and 4 contain several results labeled as 'Claim' because the authors could not prove them; Section 5 is intended to be a complete proof via Apéry sets.

Significance. If the main result is correct, it gives a neat explicit answer for an infinite family of three-generated numerical semigroups, complementing known results on consecutive squares and cubes. The Section 5 strategy is attractive: it uses exact Apéry-set descriptions verified by cardinality, with no fitted parameters, and the underlying Fibonacci identities (Lemmas 5.1, 5.6, 5.9, 5.12) are checkable and appear correct. The paper also documents errors in Tripathi's original article and provides corrected statements. The main limitation is that the Apéry-set inclusion in Propositions 5.7, 5.10, and 5.13 is not actually proved, and the two other algorithmic approaches are explicitly incomplete, so the claimed comparison does not have the rigor of a full proof.

major comments (2)
  1. [§5, Propositions 5.7, 5.10, 5.13] The proof of each of these propositions consists of the assertion 'By Lemma 5.6/5.9/5.12, we have that Ap(S(n), f_n^2) ⊆ {λ f_{n+1}^2 + μ f_{n+2}^2 | (λ,μ) ∈ C1×C2 \ C3×C4}' followed by a cardinality count. The inclusion is the load-bearing step, and it is not demonstrated. The identities in Lemma 5.6(2) and (3), for instance, reduce one coefficient while increasing the other, so it is not immediate that repeated application terminates inside the box C1×C2, nor that the excluded rectangle C3×C4 is exactly the set of pairs whose element is not the least representative of its residue class modulo f_n^2. The cardinality computation is sound but cannot compensate: equality of cardinalities gives the desired set equality only after the inclusion is independently established. Since Theorem 5.15 and Corollary 5.16 rest on these Apéry-set descriptions, the main theorem currently lacks a complete proof.
  2. [Sections 3 and 4 (Claims 3.6, 4.16, 4.18, 4.23, 4.25)] The manuscript explicitly states in the Introduction and in Section 6 that several results in Sections 3 and 4 are presented as 'Claims' for which the authors 'have not been able to provide rigorous proofs.' Consequently, the paper's title and abstract overstate the comparison: only one of the three algorithmic processes (the Rosales–García-Sánchez approach) is claimed to be fully proved, and even that proof is incomplete because of the gap described above. The remaining two processes provide consistency checks and computational evidence, not proofs. If the paper is to claim a comparison of three algorithms, the authors should either prove the Claims or clearly state that the comparison is heuristic/conditional.
minor comments (4)
  1. [§5, proof of Lemma 5.9] In the proof of Lemma 5.9, the reference to 'Expression 1 of Lemma 2.5' should be to Lemma 5.1, since Lemma 2.5 is a different statement about the embedding dimension of S(n).
  2. [§5, proof of Lemma 5.12] The phrase 'from we get Expression 1' should read 'from which we get Expression 1.'
  3. [Abstract and Section 1] The name 'Tripathy' appears in Remark 2.4 and elsewhere, while the cited author and the main text generally use 'Tripathi'; please standardize the spelling.
  4. [§5, Remark 5.3] The sentence 'However, the equality is not true as a consequence of Lemma 5.2' is unclear; Lemma 5.2 only computes the cardinality of the displayed box, and the remark should explicitly say that the box is a superset whose cardinality is larger than needed, so equality with Ap(S(n), f_n^2) does not follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main Frobenius-number derivation is self-contained; the Apéry-set inclusion is asserted without a full reduction proof, but that is a proof gap, not a circular reduction.

full rationale

The paper's central claim (Corollary 5.16) is derived in Section 5 from an explicit description of the Apéry set Ap(S(n), f_n^2). That description (Propositions 5.7, 5.10, 5.13) is built from Fibonacci-algebra identities (Lemmas 5.1, 5.6, 5.9, 5.12) plus a cardinality count: the sets C1×C2\C3×C4 are shown to have size f_n^2, and the inclusion is asserted via the identities. No parameter is fitted to the target Frobenius number, and the final formula is obtained by taking the maximum of the two pseudo-Frobenius numbers and proving which one is larger by explicit inequalities. The algorithms in Sections 3 and 4 are marked as unproved 'Claims' and are cross-checked against Section 5, but they are not used to establish the main theorem; hence they cannot smuggle the conclusion in. Self-citations to [20] and [21] are for standard Apéry-set facts and general algorithms, not for the specific formula. The only questionable step is the assertion that Lemma 5.6/5.9/5.12 imply the Apéry-set inclusion without showing the termination of the coefficient-reduction process; that is a proof gap (and correctness risk), not a circularity, since the inclusion is not assumed equal to the conclusion and no equation in the paper reduces Corollary 5.16 to a definitional input. For these reasons the paper does not exhibit any of the enumerated circularity patterns.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies only on standard mathematical background and Fibonacci identities. The unproved claims in Sections 3 and 4 are explicit conjectures, not fitted parameters.

assumptions (2)
  • standard math Standard identities for Fibonacci numbers: Cassini's identity f_{n+1}f_{n-1} - f_n² = (-1)^n and various addition formulas such as f_n = 3f_{n-3} + 2f_{n-4} and f_n < 2f_{n-1}.
    Used throughout Section 5 derivations, e.g., Lemmas 4.12, 5.1, 4.13.
  • domain assumption Basic numerical semigroup theory: Apéry set properties (Lemma 2.4 of [21], Lemma 3 of [4]), the Frobenius formula F(S) = max(Ap(S,m)) - m, and the type result t(S(n)) = 2 from Proposition 2.7 of [21].
    These background results define Ap(S,m) and link it to the Frobenius number and pseudo-Frobenius numbers.

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Pith. "Pith review of The Frobenius number corresponding to the squares of three consecutive Fibonacci numbers: comparison of three algorithmic processes." pith.science (2026). https://pith.science/paper/ZO4VV4IN

@misc{pith2026250701898,
  author       = {Pith},
  title        = {Pith review of: The Frobenius number corresponding to the squares of three consecutive Fibonacci numbers: comparison of three algorithmic processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZO4VV4IN}},
  note         = {Machine review of arXiv:2507.01898}
}
read the original abstract

We compute the Frobenius number for numerical semigroups generated by the squares of three consecutive Fibonacci numbers. We achieve this by using and comparing three distinct algorithmic approaches: those developed by Ram\'irez Alfons\'in and R{\o}dseth ([15]), Rosales and Garc\'ia-S\'anchez ([20]), and Tripathi ([26]).

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On numerical semigroups with embedding dimension four

    math.NT 2026-04 unverdicted novelty 6.0 of 10

    A geometric procedure computes Apéry sets for numerical semigroups with embedding dimension four, yielding Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees for semigroups generate...

  2. On numerical semigroups with embedding dimension four

    math.NT 2026-04 unverdicted novelty 6.0 of 10

    Develops a geometric method for the Apéry set of any numerical semigroup with embedding dimension four and illustrates it on two families of generators.

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Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper

  1. [9]

    Lepilov, J

    M. Lepilov, J. O’Rourke, and I. Swanson, Frobenius numbers of numerical semigroups generated by three consecutive squares or cubes, Semigroup Forum 91 (2015), 238–259. 17

    work page 2015

  2. [1]

    Ap´ ery, Sur les branches superlin´ eaires des courbes alg´ ebriques,C

    R. Ap´ ery, Sur les branches superlin´ eaires des courbes alg´ ebriques,C. R. Acad. Sci. Paris 222 (1946), 1198–1200

    work page 1946

  3. [2]

    Barucci, D

    V. Barucci, D. E. Dobbs, and M. Fontana, Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains , Mem. Amer. Math. Soc. 598 (1997)

    work page 1997

  4. [3]

    Brauer, On a problem of partitions, Amer

    A. Brauer, On a problem of partitions, Amer. J. Math. 64 (1942), 299–312

    work page 1942

  5. [4]

    Brauer and J

    A. Brauer and J. E. Shockley, On a problem of Frobenius, J. Reine Angew. Math. 211 (1962), 215–220

    work page 1962

  6. [5]

    Curtis, On formulas for the Frobenius number of a numerical semigroup, Math

    F. Curtis, On formulas for the Frobenius number of a numerical semigroup, Math. Scand. 67 (1990), 190–192

    work page 1990

  7. [6]

    Fr¨ oberg, G

    R. Fr¨ oberg, G. Gottlieb, and R. H¨ aggkvist, On numerical semigroups,Semigroup Forum 35 (1987), 63–83

    work page 1987

  8. [7]

    Greenberg, Solution to a Diophantine equation for nonnegative integers, J

    H. Greenberg, Solution to a Diophantine equation for nonnegative integers, J. Algorithms 9(3) (1988), 343–353

    work page 1988

Show all 26 references
  1. [8]

    S. M. Johnson, A linear Diophantine problem, Can. J. Math. 12 (1960), 390–398

    1960

  2. [10]

    Lewin, An algorithm for a solution of a problem of Frobenius, J

    M. Lewin, An algorithm for a solution of a problem of Frobenius, J. Reine Angew. Math. 276 (1975), 68–82

    1975

  3. [11]

    Moscariello, On integers which are representable as sums of large squares, Int

    A. Moscariello, On integers which are representable as sums of large squares, Int. J. Number Theory 11 (2015), 2505–2511

    2015

  4. [12]

    D. C. Ong and V. Ponomarenko, The Frobenius number of geometric sequences, Integers 8 (2008), #A33 (3 pages)

    2008

  5. [13]

    J. L. Ram ´ ırez Alfons ´ ın, Complexity of the Frobenius problem,Combinatorica 16 (1996), 143–147

    1996

  6. [14]

    J. L. Ram ´ ırez Alfons ´ ın,The Diophantine Frobenius Problem, Oxford Lectures Series in Mathematics and its Applications, vol. 30 (Oxford Univ. Press, Oxford, 2005)

    2005

  7. [15]

    J. L. Ram ´ ırez Alfons ´ ın and Ø. J. Rødseth, Numerical semigroups: Ap´ ery sets and Hilbert series, Semigroup Forum 79 (2009), 323–340

    2009

  8. [16]

    J. B. Roberts, Note on linear forms, Proc. Amer. Math. Soc. 7 (1956), 465–469

    1956

  9. [17]

    A. M. Robles-P´ erez and J. C. Rosales, The Frobenius number for sequences of triangular and tetrahedral numbers, J. Number Theory 186 (2018), 473–492

    2018

  10. [18]

    Ø. J. Rødseth, On a linear Diophantine problem of Frobenius, J. Reine Angew. Math. 301 (1978), 171–178

    1978

  11. [19]

    J. C. Rosales and M. B. Branco, Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups, J. Pure Appl. Algebra 171 (2002), 303–314

    2002

  12. [20]

    J. C. Rosales and P. A. Garc ´ ıa-S´ anchez, Numerical semigroups with embedding dimension three, Arch. Math. (Basel) , 83(6) (2004), 488–496

    2004

  13. [21]

    J. C. Rosales and P. A. Garc ´ ıa-S´ anchez,Numerical Semigroups, Developments in Mathematics, vol. 20 (Springer, New York, 2009)

    2009

  14. [22]

    E. S. Selmer, On the linear diophantine problem of Frobenius, J. Reine Angew. Math. 293/294 (1977), 1–17

    1977

  15. [23]

    Suhajda, The Frobenius number: exploring the world of non-representable integers , Bachelor’s Theses in Mathematical Sciences LUNFMA-4161-2024 (Lund University, 2024)

    P. Suhajda, The Frobenius number: exploring the world of non-representable integers , Bachelor’s Theses in Mathematical Sciences LUNFMA-4161-2024 (Lund University, 2024). http://lup.lub. lu.se/student-papers/record/9150323

    2024

  16. [24]

    The Frobenius number for sequences of triangular and tetrahedral numbers

    B. Sury, Review of the article “The Frobenius number for sequences of triangular and tetrahedral numbers”, Mathematical Reviews MR3758226 (2018)

    2018

  17. [25]

    J. J. Sylvester, Problem 7382, The Educational Times, and Journal of the College of Preceptors, New Ser. , 36(266) (1883), 177. Solution by W. J. Curran Sharp, ibidem, 36(271) (1883), 315

  18. [26]

    Tripathi, Formulae for the Frobenius number in three variables, J

    A. Tripathi, Formulae for the Frobenius number in three variables, J. Number Theory 170 (2017), 368–389. 18

    2017

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