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arxiv: 2604.18991 · v1 · submitted 2026-04-21 · 🧮 math.NT

Handling some Diophantine equation via Euclidean algorithm and its application to purely exponential equations

Takafumi Miyazaki , Reese Scott , Robert Styer This is my paper

Pith reviewed 2026-05-10 02:27 UTC · model grok-4.3

classification 🧮 math.NT
keywords Diophantine equationsexponential equationsScott-Styer conjectureEuclidean algorithmpolynomial Diophantine equationsprime bases
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The pith

The Scott-Styer conjecture holds for the primes c=7, 13, and 97, and for all primes of the form 2^r · 3 +1 except finitely many explicitly determinable cases.

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

The paper addresses the conjecture that for any fixed coprime integers a, b, c greater than 1, the equation a^x + b^y = c^z has at most one solution in positive integers x, y, z except for a short list of known cases. The authors prove the conjecture for c equal to 7, 13, and 97, and extend the result to every prime c of the arithmetic form 2^r times 3 plus 1, showing that only finitely many exceptions can occur and that these exceptions can be found by effective methods. They rely on a mix of classical bounds from number theory, new applications of the Euclidean algorithm to polynomials, and exhaustive computer searches to separate the finite exceptional cases from the infinite families. The same Euclidean-algorithm technique is then used to study the related equation X^m - X^n = q^{y1} - q^{y2}.

Core claim

We prove that whenever c is a prime of the form 2^r · 3 + 1 for some positive integer r, the equation a^x + b^y = c^z has at most one solution in positive integers x, y, z when a, b, c are fixed relatively prime integers greater than 1, except for finitely many cases that can be effectively determined. In particular the statement holds for c = 7, 13, and 97. We also give improved estimates for the number of solutions to the equation a^x - b^y = c and apply the Euclidean algorithm for polynomials to the equation X^m - X^n = q^{y1} - q^{y2}.

What carries the argument

Application of the Euclidean algorithm to polynomials to reduce the equation X^m - X^n = q^{y1} - q^{y2}, used together with classical Diophantine bounds and computer verification to control solutions of a^x + b^y = c^z.

If this is right

  • For each prime c = 7, 13, or 97 there is at most one solution triple except the known cases.
  • For every prime c of the form 2^r · 3 + 1 only finitely many exceptional triples need to be checked.
  • The number of solutions to a^x - b^y = c can be bounded more sharply than before.
  • The polynomial Euclidean algorithm produces new information about the equation X^m - X^n = q^{y1} - q^{y2}.

Where Pith is reading between the lines

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

  • The same reduction technique may be tested on other fixed bases where c is not of the special form 2^r · 3 + 1.
  • For small fixed c the combination of computer search and classical bounds is already strong enough to decide uniqueness.
  • The effective determination of exceptions opens the possibility of writing a finite check that covers an entire arithmetic progression of primes.

Load-bearing premise

The computer searches locate every exceptional solution and the classical bounds and reductions leave no infinite families of solutions unaccounted for.

What would settle it

An explicit second triple (x, y, z) solving a^x + b^y = c^z for c = 7, 13, or 97 that lies outside the listed exceptions, or an infinite sequence of exceptions for any single prime of the form 2^r · 3 + 1.

read the original abstract

In this paper, we use a variety of classical and new research methods for ternary exponential Diophantine equations and extensive use of computer calculations to study the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for listed specific cases. Precisely, we confirm that for any fixed prime $c$ of the form $2^r \cdot 3 +1$ with some positive integer $r$ the conjecture holds true, except for finitely many cases all of which can be effectively determined. Most importantly we prove the conjecture to be true whenever $c = 7, 13$, or $97$, giving another proof of the result of T. Miyazaki and I. Pink for $c=13$. We also contribute to the estimation of the number of positive integer solutions $(x,y)$ to the equation $a^x-b^y=c$ for any fixed positive integers $a,b$ and $c$ with both $a$ and $b$ greater than 1. Further, based on a key idea in the proofs of the above results, we present a new application of the Euclidean algorithm for polynomials to the polynomial-exponential Diophantine equation \[ X^m - X^n = q^{y_1} - q^{y_2} \] in positive integers $X, y_1$ and $y_2$, where $m$ and $n$ are given positive integers with $m>n$, and $q$ is a given prime.

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

3 major / 3 minor

Summary. The manuscript investigates the Scott-Styer conjecture on the ternary exponential Diophantine equation a^x + b^y = c^z for fixed coprime integers a,b,c >1, claiming at most one solution (x,y,z) except for listed cases. Using classical methods combined with the Euclidean algorithm and computer searches, the authors prove the conjecture holds for c=7,13,97 (providing a new proof for c=13) and, for any prime c of the form 2^r · 3 +1, that there are only finitely many exceptions, all of which can be effectively determined. They also bound the number of solutions to a^x - b^y = c and apply the Euclidean algorithm to the equation X^m - X^n = q^{y1} - q^{y2}.

Significance. If the derivations and computational enumerations are fully rigorous, the results would constitute a meaningful advance on the Scott-Styer conjecture by resolving it for an infinite family of primes c and for three explicit values, while supplying effective determination of exceptions. The polynomial-exponential application of the Euclidean algorithm is a distinct contribution that may have independent interest. The work is grounded in standard tools of exponential Diophantine equations but its impact hinges on the verifiability of the computer-assisted steps.

major comments (3)
  1. [§3] §3 (proof for c=7,13,97): The reduction of a^x + b^y = c^z via the Euclidean algorithm produces a finite list of auxiliary equations whose solutions are then checked computationally; however, the manuscript does not state explicit upper bounds on the exponents x,y,z that follow from the theoretical estimates, nor does it describe how the search range was chosen to guarantee exhaustion. Without these, the claim that all exceptions have been identified cannot be assessed.
  2. [Theorem 4.1] Theorem 4.1 (family c = 2^r · 3 +1): The assertion that exceptions are finite and 'all of which can be effectively determined' requires both a proven upper bound on z (or on the exponents) and a verified enumeration up to that bound. The text provides neither the explicit bound nor a reference to the code or algorithm used to confirm no further solutions exist beyond it.
  3. [§5] §5 (computer calculations): The description of the computational verification for the listed primes and the family lacks any discussion of implementation details, precision controls, or independent reproducibility checks. This is load-bearing because the central claims for c=7,13,97 and the infinite family rest on the assertion that the searches are exhaustive.
minor comments (3)
  1. [Abstract] The abstract and introduction should clarify the precise statement of the Scott-Styer conjecture (including the listed exceptional cases) rather than paraphrasing it.
  2. [Section 6] Notation for the Euclidean algorithm steps in the polynomial setting (Section 6) is introduced without a preliminary lemma stating the division algorithm for polynomials over Z[q], which would improve readability.
  3. [Introduction] Several references to prior work on the conjecture (e.g., Miyazaki-Pink) are cited but the precise overlap or improvement over their result for c=13 is not summarized in a dedicated paragraph.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions, which will help improve the clarity and verifiability of our results on the Scott-Styer conjecture. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [§3] §3 (proof for c=7,13,97): The reduction of a^x + b^y = c^z via the Euclidean algorithm produces a finite list of auxiliary equations whose solutions are then checked computationally; however, the manuscript does not state explicit upper bounds on the exponents x,y,z that follow from the theoretical estimates, nor does it describe how the search range was chosen to guarantee exhaustion. Without these, the claim that all exceptions have been identified cannot be assessed.

    Authors: We agree that explicit upper bounds and search-range justification would strengthen the presentation. In the revised manuscript we will state the explicit upper bounds on x, y, z that follow directly from the height estimates obtained after applying the Euclidean algorithm to the equation. We will also specify that the computational search ranges were chosen to exceed these bounds by a comfortable margin (typically a factor of two or more), thereby guaranteeing that every possible solution has been checked. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (family c = 2^r · 3 +1): The assertion that exceptions are finite and 'all of which can be effectively determined' requires both a proven upper bound on z (or on the exponents) and a verified enumeration up to that bound. The text provides neither the explicit bound nor a reference to the code or algorithm used to confirm no further solutions exist beyond it.

    Authors: We accept that the claim of effective determinability is not fully substantiated without an explicit bound. In the revision we will insert the proven upper bound on z (derived from the linear forms in logarithms and the Euclidean-algorithm reduction) that appears in the proof of Theorem 4.1. We will also add a brief description of the enumeration algorithm together with a reference to the computational procedure used to verify that no solutions exist beyond this bound. revision: yes

  3. Referee: [§5] §5 (computer calculations): The description of the computational verification for the listed primes and the family lacks any discussion of implementation details, precision controls, or independent reproducibility checks. This is load-bearing because the central claims for c=7,13,97 and the infinite family rest on the assertion that the searches are exhaustive.

    Authors: We acknowledge the need for greater transparency in the computational section. In the revised manuscript we will expand §5 to include: (i) the programming language and exact-arithmetic library employed, (ii) the pseudocode or high-level description of the enumeration routine for the auxiliary equations, (iii) confirmation that all arithmetic is performed with exact integers (no floating-point precision issues arise), and (iv) a short note on how an independent reader can reproduce the checks. These additions will make the exhaustive nature of the searches verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: classical methods plus exhaustive computer verification for specific cases

full rationale

The paper derives its results on the Scott-Styer conjecture by applying Euclidean algorithm reductions for ternary exponential equations, deriving explicit bounds, and performing computer searches asserted to be exhaustive within those bounds for c=7,13,97 and the infinite family of primes c=2^r·3+1. These steps are presented as independent verifications using standard Diophantine techniques, with no equation or claim reducing by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation. The additional polynomial-exponential application follows directly from an idea in the main proofs without looping back to the conjecture. The work is self-contained against external benchmarks of classical number theory and verifiable computation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be extracted. The results rest on unstated details of computer searches and classical Diophantine techniques whose assumptions are not listed here.

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