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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 4

A note on Terai's conjecture concerning primitive Pythagorean triples

Maohua Le(Lingnan Normal College)0000-0002-7502-2496
Gökhan Soydan(BURSA ULUDAG UNIVERSITY)0000-0002-6321-4132
Sayfa
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Yayın Tarihi
2021-08-06
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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 $2\nmid s$, $r\ge 2$ and $s<2^{r-1}$, then Terai's conjecture is true.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Let $f,g$ be positive integers such that $f>g$, $\gcd(f,g)=1$ and $f\not\equiv g \pmod{2}$.
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Temel Bulgular & Veri: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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Sonuç & Katkı: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 $2\nmid s$, $r\ge 2$ and $s<2^{r-1}$, then Terai's conjecture is true.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:polynomial-exponential Diophantine equation · generalized Ramanujan-Nagell equation · primitive Pythagorean triple

Abstract

Let $f,g$ be positive integers such that $f>g$, $\gcd(f,g)=1$ and $f\not\equiv g \pmod{2}$. 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 $2\nmid s$, $r\ge 2$ and $s<2^{r-1}$, then Terai's conjecture is true.

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Bu Makaleye Atıf Yap

Le, M., Soydan, G. (2021). A note on Terai's conjecture concerning primitive Pythagorean triples. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 4. https://doi.org/10.15672/hujms.795889

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Kaynakça (13)

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