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arxiv: 2502.00296 · v2 · submitted 2025-02-01 · 🧮 math.NT

Representing an integer and its powers in two unrelated number systems

Divyum Sharma , L. Singhal This is my paper

Pith reviewed 2026-05-23 03:49 UTC · model grok-4.3

classification 🧮 math.NT
keywords Diophantine equationHamming weightZeckendorf representationconvergentsquadratic irrationalradix representationupper bounds
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The pith

Effective upper bounds exist for y^a expressed as a sum of convergent denominators to a quadratic irrational, based on the Hamming weights of y in radix and Zeckendorf representations.

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

The paper examines Diophantine equations in which a power y to the a equals a finite sum of terms from the sequence of denominators of the convergents to a fixed quadratic irrational alpha. It derives two explicit upper bounds on the size of such y^a, one depending on the number of nonzero digits of y in a fixed radix base and the other on its Zeckendorf representation. The Zeckendorf bound extends an earlier result of Vukusic and Ziegler. These bounds arise because the convergent sequence inherits strong additive relations from the continued fraction expansion of a quadratic irrational, allowing control over how many terms are needed to reach a given size.

Core claim

If y^a equals a sum of K terms q_{N_i} from the convergent denominators to alpha, then y^a is bounded above by a quantity depending on the Hamming weight of y with respect to its radix representation, and separately by a quantity depending on the Hamming weight with respect to its Zeckendorf representation. An analogue of the Kebli-Kihel-Larone-Luca theorem is also obtained in this setting.

What carries the argument

The sequence of convergent denominators (q_N) to the quadratic irrational alpha, which satisfies recurrence relations allowing the Hamming weight arguments to bound the sum.

If this is right

  • The size of y^a is effectively limited by the sparsity of its digit expansions in the two systems.
  • Only finitely many solutions exist for fixed Hamming weight and a.
  • The extension of the Vukusic-Ziegler result applies directly to this equation.
  • An analogue theorem holds for the representation in this number system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the quadratic irrational has bounded partial quotients, the bounds may become particularly sharp.
  • Similar techniques could apply to other sequences defined by linear recurrences beyond convergents.
  • Computational searches for solutions could be restricted to y with small Hamming weight.

Load-bearing premise

The convergent denominator sequence must obey the precise additive and recurrence relations that follow from alpha being a quadratic irrational.

What would settle it

An explicit counterexample consisting of a quadratic irrational alpha, integers y and a at least 2, and a sum of its convergent denominators equaling y^a where the value exceeds both the radix-based and Zeckendorf-based upper bounds derived in the paper.

Figures

Figures reproduced from arXiv: 2502.00296 by Divyum Sharma, L. Singhal.

Figure 1
Figure 1. Figure 1: Intermediate bounds to be computed [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original abstract

Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent denominators to $\alpha$. We find two effective upper bounds for $y^a$ which depend on the Hamming weights of $y$ with respect to its radix and Zeckendorf representations, respectively. The latter bound extends a recent result of Vukusic and Ziegler. En route, we obtain an analogue of a theorem by Kebli, Kihel, Larone and Luca.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the Diophantine equation y^a = q_{N_1} + ⋯ + q_{N_K} (N_1 ≥ ⋯ ≥ N_K ≥ 0, a,y ≥ 2) where (q_N) denotes the sequence of convergent denominators to a fixed quadratic irrational α. It derives two effective upper bounds on y^a, one in terms of the radix-b Hamming weight of y and one in terms of the Zeckendorf Hamming weight of y. The Zeckendorf bound extends a recent result of Vukusic and Ziegler; an analogue of a theorem of Kebli–Kihel–Larone–Luca is obtained along the way.

Significance. If the derivations are correct, the work supplies effective bounds that can be used to limit the search space for solutions of superelliptic equations whose right-hand sides are sums of convergents. The explicit dependence on Hamming weights in two unrelated positional systems is a clear strengthening of the Vukusic–Ziegler result, and the analogue of Kebli et al. broadens the applicability of the method to a wider class of linear recurrence sequences.

major comments (2)
  1. [§3] §3, proof of Theorem 3.1: the passage from the radix-Hamming-weight bound to the stated explicit constant requires that the partial sums of at most K terms of (q_N) satisfy a uniform additive estimate; the manuscript invokes the standard recurrence q_{n+2}=a q_{n+1}+q_n but does not record the dependence of the implied constant on K or on the continued-fraction partial quotients of α.
  2. [Theorem 4.3] Theorem 4.3 (Zeckendorf bound): the reduction to the Vukusic–Ziegler setting uses that every sum of distinct q_N can be rewritten with a bounded number of carries in the Zeckendorf representation; the carry bound is stated to be independent of K, but the argument only controls carries when the indices differ by at least 2, leaving the case of consecutive indices unexamined.
minor comments (2)
  1. [Introduction] The statement of the main theorems should explicitly record that α is fixed (so that all implied constants may depend on α) rather than leaving this implicit in the phrase “fixed quadratic irrational.”
  2. [§2] Notation: the symbol wt_b(y) for the radix-b Hamming weight is introduced without a preceding definition; a short sentence in §2 would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments, which help clarify the presentation of the effective bounds. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3, proof of Theorem 3.1: the passage from the radix-Hamming-weight bound to the stated explicit constant requires that the partial sums of at most K terms of (q_N) satisfy a uniform additive estimate; the manuscript invokes the standard recurrence q_{n+2}=a q_{n+1}+q_n but does not record the dependence of the implied constant on K or on the continued-fraction partial quotients of α.

    Authors: We agree that the dependence on K and the partial quotients of α must be recorded explicitly. Since α is fixed, its partial quotients are bounded by a constant A_α. In the revision we will derive and state the explicit form of the additive constant in terms of K and A_α, obtained by iterating the recurrence at most K times. This makes the bound fully effective and transparent while leaving the main theorem unchanged. revision: yes

  2. Referee: [Theorem 4.3] Theorem 4.3 (Zeckendorf bound): the reduction to the Vukusic–Ziegler setting uses that every sum of distinct q_N can be rewritten with a bounded number of carries in the Zeckendorf representation; the carry bound is stated to be independent of K, but the argument only controls carries when the indices differ by at least 2, leaving the case of consecutive indices unexamined.

    Authors: The referee correctly identifies that the carry analysis for consecutive indices was not treated separately. We will revise the proof of Theorem 4.3 by adding a short case analysis for runs of consecutive indices. Because the sequence (q_N) satisfies a linear recurrence of order 2, any carry propagation initiated by consecutive terms remains bounded by a constant depending only on the recurrence coefficients (hence independent of K). The revised argument will therefore confirm that the overall carry bound is independent of K. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on the standard recurrence and limited additivity properties of convergent denominators to a fixed quadratic irrational, which are external facts about such sequences and not constructed within the paper. The two effective upper bounds on y^a are obtained by applying Hamming-weight arguments to these properties, extending results of Vukusic-Ziegler and Kebli et al. (distinct authors) without any reduction of the claimed bounds to fitted inputs, self-definitions, or load-bearing self-citations. The central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.0 · 5660 in / 1020 out tokens · 67301 ms · 2026-05-23T03:49:42.475945+00:00 · methodology

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    On perfect powers that are sums of two Pell numbers

    Hyacinthe Aboudja, Mohand Hernane, Salah Eddine Rihane , and Alain Togb´ e. On perfect powers that are sums of two Pell numbers. Period. Math. Hungar. , 82(1):11–15, 2021

    work page 2021

  2. [2]

    Automatic sequences

    Jean-Paul Allouche and Jeffrey Shallit. Automatic sequences . Cambridge University Press, Cambridge, 2003. Theory, applications, generalizations

    work page 2003

  3. [3]

    On the resolution of the diophantine equation Un + Um = xq

    Pritam Kumar Bhoi, Sudhansu Sekhar Rout, and Gopal Krish na Panda. On the resolution of the diophantine equation Un + Um = xq. arXiv:2202.11934, 2022

    work page arXiv 2022

  4. [4]

    Bravo and Florian Luca

    Jhon J. Bravo and Florian Luca. On the Diophantine equati on Fn + Fm = 2 a. Quaest. Math. , 39(3):391–400, 2016

    work page 2016

  5. [5]

    On the re presentation of Fibonacci and Lucas numbers in an integer base

    Yann Bugeaud, Mihai Cipu, and Maurice Mignotte. On the re presentation of Fibonacci and Lucas numbers in an integer base. Ann. Math. Qu´ e., 37(1):31–43, 2013

    work page 2013

  6. [6]

    Class ical and modular approaches to exponential Diophantine equations

    Yann Bugeaud, Maurice Mignotte, and Samir Siksek. Class ical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers. Ann. of Math. (2), 163(3):969–1018, 2006

    work page 2006

  7. [7]

    M iller, Carsten Peterson, and Yen Nhi Truong Vu

    Katherine Cordwell, Max Hlavacek, Chi Huynh, Steven J. M iller, Carsten Peterson, and Yen Nhi Truong Vu. Summand minimality and asymptotic convergen ce of generalized Zeckendorf decompositions. Res. Number Theory , 4(4):27 pp., 2018

    work page 2018

  8. [8]

    Tiebekabe

    Benjamin Earp-Lynch, Simon Earp-Lynch, Omar Kihel, and P. Tiebekabe. Powers as fi- bonacci sums. Quaest. Math. , pages 1–13, 2024

    work page 2024

  9. [9]

    On the nonnegative integer solutions to the equation Fn ± Fm = ya

    Salima Kebli, Omar Kihel, Jesse Larone, and Florian Luca . On the nonnegative integer solutions to the equation Fn ± Fm = ya. J. Number Theory , 220:107–127, 2021

    work page 2021

  10. [10]

    On the nonnegative integer solutions of the equation Fn±Fm = ya

    Omar Kihel and Jesse Larone. On the nonnegative integer solutions of the equation Fn±Fm = ya. Quaest. Math. , 44(8):1133–1139, 2021

    work page 2021

  11. [11]

    H. W. Lenstra, Jr. and J. O. Shallit. Continued fraction s and linear recurrences. Math. Comp., 61(203):351–354, 1993

    work page 1993

  12. [12]

    On perfect powers that a re sums of two Fibonacci numbers

    Florian Luca and Vandita Patel. On perfect powers that a re sums of two Fibonacci numbers. J. Number Theory , 189:90–96, 2018

    work page 2018

  13. [13]

    E. M. Matveev. An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. Izv. Ross. Akad. Nauk Ser. Mat. , 62(4):81–136, 1998

    work page 1998

  14. [14]

    E. M. Matveev. An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Ross. Akad. Nauk Ser. Mat. , 64(6):125–180, 2000

    work page 2000

  15. [15]

    Bemerkungen zur Theorie der Diop hantischen Approximationen

    Alexander Ostrowski. Bemerkungen zur Theorie der Diop hantischen Approximationen. Abh. Math. Sem. Univ. Hamburg , 1(1):77–98, 1922

    work page 1922

  16. [16]

    Peth¨ o.Perfect powers in second order recurrences , volume 34 of Colloq

    A. Peth¨ o.Perfect powers in second order recurrences , volume 34 of Colloq. Math. Soc. J´ anos Bolyai. North-Holland, Amsterdam, 1984

    work page 1984

  17. [17]

    Peth¨ o and B

    A. Peth¨ o and B. M. M. de W eger. Products of prime powers i n binary recurrence sequences. I. The hyperbolic case, with an application to the generaliz ed Ramanujan-Nagell equation. Mathematics of Computation , 47(176):713–727, 1986

    work page 1986

  18. [18]

    Perfect powers in second order linear re currences

    Attila Peth¨ o. Perfect powers in second order linear re currences. J. Number Theory , 15(1):5– 13, 1982

    work page 1982

  19. [19]

    On sums of two Fibona cci numbers that are powers of numbers with limited Hamming weight

    Ingrid Vukusic and Volker Ziegler. On sums of two Fibona cci numbers that are powers of numbers with limited Hamming weight. Quaest. Math. , 47(4):851–869, 2024

    work page 2024

  20. [20]

    Diophantine approximation on linear algebraic groups , volume 326 of Grundlehren der mathematischen Wissenschaften [Fundamen tal Principles of Mathematical Sciences]

    Michel W aldschmidt. Diophantine approximation on linear algebraic groups , volume 326 of Grundlehren der mathematischen Wissenschaften [Fundamen tal Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000. Transcendence propertie s of the exponential function in several variables

    work page 2000

  21. [21]

    Zeckendorf

    E. Zeckendorf. Repr´ esentation des nombres naturels p ar une somme de nombres de fibonacci ou de nombres de lucas. Bull. Soc. Roy. Sci. Li` ege , 41:179–182, 1972. PERFECT POWERS AS SUMS OF CONVERGENT DENOMINATORS 17

    work page 1972

  22. [22]

    Effective results for linear equations in members of two recurrence sequences

    Volker Ziegler. Effective results for linear equations in members of two recurrence sequences. Acta Arith., 190(2):139–169, 2019

    work page 2019

  23. [23]

    Sums of Fibonacci numbers that are perf ect powers

    Volker Ziegler. Sums of Fibonacci numbers that are perf ect powers. Quaest. Math. , 46(8):1717–1742, 2023. Department of Mathematics, Birla Institute of Technology a nd Science, Pilani 333 031 India Email address : divyum.sharma@pilani.bits-pilani.ac.in Apni Manzil, Nai Sadak, Sajjanon ki Dhani, Kajara 333 030 India Email address : lovysinghal@gmail.com

    work page 2023