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General sharp bounds for the number of solutions to purely exponential equations with three terms

Maohua Le, Takafumi Miyazaki

TL;DR

This work advances sharp, general bounds for the number of solutions to the purely exponential equation $a^x+b^y=c^z$ with fixed coprime integers $a,b,c>1$, showing that for any fixed $c$ there is at most one solution except for finitely many $(a,b)$ pairs. The authors build on p-adic methods from Miyazaki–Pink and Ridout’s theorem to reduce the problem to a concrete elementary system and then solve it completely, yielding explicit exceptional cases $(a,b,c,z,Z)=(2,5,3,2,3)$ and $(5,2,3,3,2)$. They also develop strategies toward an effective version of the result for certain $c$, including a comprehensive discussion of rational approximation of algebraic irrationals, a sieve-based restriction of unknowns, and Baker-type bounds to bound exponents. The work unifies previous finiteness results, provides a clear path to effective determination of exceptional pairs for many $c$, and highlights several open problems that, if resolved, could render the main theorem effective for infinitely many values of $c$ with substantial computational content.

Abstract

It is conjectured 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 specific cases. In this paper, we prove that for any fixed $c$ there is at most one solution to the equation, except for only finitely many pairs of $a$ and $b.$ This is regarded as a 3-variable generalization of the result of Miyazaki and Pink [T. Miyazaki and I. Pink, Number of solutions to a special type of unit equations in two unknowns, III, arXiv:2403.20037 (accepted for publication in Math. Proc. Cambridge Philos. Soc.)] which asserts that for any fixed positive integer $a$ there are only finitely many pairs of coprime positive integers $b$ and $c$ with $b>1$ such that the Pillai's type equation $a^x-b^y=c$ has more than one solution in positive integers $x$ and $y$. The proof of our result is based on a certain $p$-adic idea of Miyazaki and Pink and relies on many deep theorems on the theory of Diophantine approximation, and it also includes the complete description of solutions to some interesting system of simultaneous polynomial-exponential equations. We also discuss how effectively exceptional pairs of $a$ and $b$ on our result for each $c$ can be determined.

General sharp bounds for the number of solutions to purely exponential equations with three terms

TL;DR

This work advances sharp, general bounds for the number of solutions to the purely exponential equation with fixed coprime integers , showing that for any fixed there is at most one solution except for finitely many pairs. The authors build on p-adic methods from Miyazaki–Pink and Ridout’s theorem to reduce the problem to a concrete elementary system and then solve it completely, yielding explicit exceptional cases and . They also develop strategies toward an effective version of the result for certain , including a comprehensive discussion of rational approximation of algebraic irrationals, a sieve-based restriction of unknowns, and Baker-type bounds to bound exponents. The work unifies previous finiteness results, provides a clear path to effective determination of exceptional pairs for many , and highlights several open problems that, if resolved, could render the main theorem effective for infinitely many values of with substantial computational content.

Abstract

It is conjectured that for any fixed relatively prime positive integers and all greater than 1 there is at most one solution to the equation in positive integers and , except for specific cases. In this paper, we prove that for any fixed there is at most one solution to the equation, except for only finitely many pairs of and This is regarded as a 3-variable generalization of the result of Miyazaki and Pink [T. Miyazaki and I. Pink, Number of solutions to a special type of unit equations in two unknowns, III, arXiv:2403.20037 (accepted for publication in Math. Proc. Cambridge Philos. Soc.)] which asserts that for any fixed positive integer there are only finitely many pairs of coprime positive integers and with such that the Pillai's type equation has more than one solution in positive integers and . The proof of our result is based on a certain -adic idea of Miyazaki and Pink and relies on many deep theorems on the theory of Diophantine approximation, and it also includes the complete description of solutions to some interesting system of simultaneous polynomial-exponential equations. We also discuss how effectively exceptional pairs of and on our result for each can be determined.
Paper Structure (18 sections, 20 theorems, 144 equations)

This paper contains 18 sections, 20 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Works of Miyazaki and Pink
  3. Proof of Theorem \ref{['th-anyc']} for main case
  4. Proof of Theorem \ref{['th-x2y1X1Y2']}
  5. Effective version of Theorem \ref{['th-anyc']} for certain cases
  6. Towards effective version of Theorem \ref{['th-anyc']} : rational approximations of algebraic irrationals
  7. Case where $X=1$
  8. Case where $X=1$ and $x \ge 2$
  9. Case where $X=1$ and $x=1$
  10. Case where $Y=1$
  11. Towards effective version of Theorem \ref{['th-anyc']} : restricting possible values of unknowns
  12. Restricting possible values of $x$
  13. Application of Proposition $\ref{['prop-x-bounds']}$
  14. Case where $c$ is prime
  15. Finding upper bounds for $x$ and $\max\{X,Y\}$
  16. ...and 3 more sections

Key Result

Proposition 1.1

There are in general at most two solutions to equation $abc,$ except for $(a,b,c)=(3,5,2)$ or $(5,3,2)$ each of which gives exactly three solutions.

Theorems & Definitions (38)

  • Proposition 1.1
  • Conjecture 1
  • Theorem 1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.1
  • ...and 28 more