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Journals / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 4

A note on Terai's conjecture concerning primitive Pythagorean triples

Maohua Le
Gökhan Soydan
Pages
911–917
Publication Date
2022-10-18
DOI
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

In this paper, using elementary number theory methods with some known results on higher Diophantine equations, we prove that if $f=2^rs$ and $g=1$, where $r,s$ are positive integers satisfying $2nmid s$, $rge 2$ and $s

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let $f,g$ be positive integers such that $f>g$, $gcd(f,g)=1$ and $fnotequiv g pmod{2}$.
📊
Key Findings & Data:Terai conjectured that the equation $x^2+(f^2-g^2)^y=(f^2+g^2)^z$ has only one positive integer solution $(x,y,z)=(2fg,2,2)$.
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Conclusion & Impact:In this paper, using elementary number theory methods with some known results on higher Diophantine equations, we prove that if $f=2^rs$ and $g=1$, where $r,s$ are positive integers satisfying $2nmid s$, $rge 2$ and $s
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

Let $f,g$ be positive integers such that $f>g$, $gcd(f,g)=1$ and $fnotequiv g pmod{2}$. In 1993, N. Terai conjectured that the equation $x^2+(f^2-g^2)^y=(f^2+g^2)^z$ has only one positive integer solution $(x,y,z)=(2fg,2,2)$. This is a problem that has not been solved yet. In this paper, using elementary number theory methods with some known results on higher Diophantine equations, we prove that if $f=2^rs$ and $g=1$, where $r,s$ are positive integers satisfying $2nmid s$, $rge 2$ and $s

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Cite This Article

Le, M., Soydan, G. (2022). A note on Terai's conjecture concerning primitive Pythagorean triples. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 4, 911–917. https://ajindex.com/articles/529440

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