[Paper Review] A Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y = c^z$

LE Mao-hua, Reese Scott+1 more|arXiv (Cornell University)|Aug 20, 2018

Algebraic Geometry and Number Theory85 references8 citations

TL;DR

This survey comprehensively reviews the ternary purely exponential Diophantine equation $a^x + b^y = c^z$ for fixed coprime positive integers $a$, $b$, $c$ with $\min\{a,b,c\} > 1$. It synthesizes recent advances on bounds for solutions, the Jeśmanowicz and Terai-Jeśmanowicz conjectures, and solution uniqueness, establishing that $N(a,b,c) \leq 2$ when $c \equiv 1 \pmod{2}$, and $N(a,b,c) \leq 3$ for large $\max\{a,b,c\}$, with further refinements under specific conditions.

ABSTRACT

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$.

Motivation & Objective

• To synthesize and organize existing research on the ternary purely exponential Diophantine equation $a^x + b^y = c^z$ for fixed coprime bases $a$, $b$, $c$ with $\min\{a,b,c\} > 1$.
• To clarify and unify the formulation of the Terai-Jeśmanowicz conjecture and its relationship with the Jeśmanowicz conjecture.
• To provide updated, effective upper bounds on the number of solutions $N(a,b,c)$, particularly under parity class analysis and logarithmic bounds.
• To present recent results showing $N(a,b,c) \leq 2$ when $c \equiv 1 \pmod{2}$, and $N(a,b,c) \leq 3$ for $\max\{a,b,c\} > 5 \times 10^{27}$, with stronger bounds under specific conditions.
• To highlight open problems and conjectures, including Miyazaki's shuffle variant of the Terai-Jeśmanowicz conjecture, and to guide future research through a structured overview.

Proposed method

• Application of $p$-adic logarithmic methods and linear forms in logarithms to derive effective upper bounds on $\max\{x,y,z\}$, improving prior results to $\max\{x,y,z\} < 6500(\log(\max\{a,b,c\}))^3$.
• Use of parity class decomposition (I–IV) to analyze solution structure, enabling bounds on the number of solutions per class.
• Leveraging the effective bound $z < \frac{1}{2}ab$ for $c \equiv 1 \pmod{2}$ to constrain solution space and reduce solution count.
• Combining elementary methods with logarithmic bounds to show $N(a,b,c) \leq 3$ when $\max\{a,b,c\} > 5 \times 10^{27}$, and $N(a,b,c) \leq 2$ when $2 \mid c$ and $\max\{a,b,c\} \geq 10^{62}$.
• Systematic review and comparison of known results on special families (e.g., Pythagorean triples, Fibonacci sequences, parametric families) to identify solution uniqueness.

Experimental results

Research questions

• RQ1 What are the current best effective upper bounds on the number of solutions $(x,y,z)$ to $a^x + b^y = c^z$ for fixed coprime $a,b,c$ with $\min\{a,b,c\} > 1$?
• RQ2 Under what conditions does the equation $a^x + b^y = c^z$ have at most one solution, and how does this relate to the Terai-Jeśmanowicz and Jeśmanowicz conjectures?
• RQ3 Can the number of solutions be bounded independently of $c$ when $c \equiv 1 \pmod{2}$, and what is the best known such bound?
• RQ4 How do parametric families (e.g., $a=f$, $b=3f^2-1$, $c=4f^2-1$) and sequences (e.g., Fibonacci and generalized Fibonacci numbers) affect solution uniqueness?
• RQ5 Is Miyazaki's shuffle variant of the Terai-Jeśmanowicz conjecture true in general, and what progress has been made toward its proof?

Key findings

• For $c \equiv 1 \pmod{2}$, the number of solutions $N(a,b,c)$ is at most 2, as shown by Scott and Styer, improving earlier bounds of $2^{\omega(c)+1}$.
• The bound $\max\{x,y,z\} < 6500(\log(\max\{a,b,c\}))^3$ was established by Hu and Le, representing a significant improvement over previous estimates.
• When $\max\{a,b,c\} > 5 \times 10^{27}$, the number of solutions $N(a,b,c)$ is at most 3, and when $2 \mid c$ and $\max\{a,b,c\} \geq 10^{62}$, $N(a,b,c) \leq 2$, under specific conditions.
• For the parametric family $(a,b,c) = (f, 3f^2-1, 4f^2-1)$, the equation has a unique solution $(x,y,z) = (2,1,1)$, confirmed independently by Chen and He-Togbé.
• For the Fibonacci-based case $(a,b,c) = (F_k, F_{k+1}, F_{2k+1})$ with $k \geq 3$, Terai's conjecture was confirmed by Miyazaki: only one solution $(x,y,z) = (2,2,1)$ exists.
• Miyazaki proved that his shuffle variant of the Terai-Jeśmanowicz conjecture holds when $q = r = 2$ and $b+1 = c$, showing that $c^X + b^Y = a^Z$ has only the solution $(X,Y,Z) = (1,1,p)$ under these conditions.