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Journals / Turkish Journal of Mathematics / 2022 / Cilt: 46 - Sayı: 4

A parametric family of ternary purely exponential Diophantine equation $A^x + B^y = C^z$

Yasutsugu FUJITA
Maohua LE
Pages
1224–1232
Publication Date
2022-11-01
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In this paper, using the Baker method, we prove that if m > $max{10^8 , c^2 }$, then the equation $(am^2 + 1)x + (bm^2 − 1)^y = (cm)^z$ has only one positive integer solution (x, y, z) = (1, 1, 2).

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let a, b, c be fixed positive integers such that $a + b = c^2 , 2 ∤ c and (b/p) ̸= 1$ for every prime divisor p of c , where (b/p) is the Legendre symbol.
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Key Findings & Data:Further let m be a positive integer with m > 1.
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Conclusion & Impact:In this paper, using the Baker method, we prove that if m > $max{10^8 , c^2 }$, then the equation $(am^2 + 1)x + (bm^2 − 1)^y = (cm)^z$ has only one positive integer solution (x, y, z) = (1, 1, 2).
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

Let a, b, c be fixed positive integers such that $a + b = c^2 , 2 ∤ c and (b/p) ̸= 1$ for every prime divisor p of c , where (b/p) is the Legendre symbol. Further let m be a positive integer with m > 1. In this paper, using the Baker method, we prove that if m > $max{10^8 , c^2 }$, then the equation $(am^2 + 1)x + (bm^2 − 1)^y = (cm)^z$ has only one positive integer solution (x, y, z) = (1, 1, 2).

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Cite This Article

FUJITA, Y., LE, M. (2022). A parametric family of ternary purely exponential Diophantine equation $A^x + B^y = C^z$. Turkish Journal of Mathematics, Cilt: 46 - Sayı: 4, 1224–1232. https://ajindex.com/articles/604276

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