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https://doi.org/10.3336/gm.55.2.03

A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\)

Maohua Le ; Institute of Mathematics, Lingnan Normal College, Guangdong, 524048 Zhangjiang, China
Gökhan Soydan   ORCID icon orcid.org/0000-0002-6321-4132 ; Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey

Puni tekst: engleski, pdf (105 KB) str. 195-201 preuzimanja: 124* citiraj
APA 6th Edition
Le, M. i Soydan, G. (2020). A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\). Glasnik matematički, 55 (2), 195-201. https://doi.org/10.3336/gm.55.2.03
MLA 8th Edition
Le, Maohua i Gökhan Soydan. "A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\)." Glasnik matematički, vol. 55, br. 2, 2020, str. 195-201. https://doi.org/10.3336/gm.55.2.03. Citirano 19.10.2021.
Chicago 17th Edition
Le, Maohua i Gökhan Soydan. "A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\)." Glasnik matematički 55, br. 2 (2020): 195-201. https://doi.org/10.3336/gm.55.2.03
Harvard
Le, M., i Soydan, G. (2020). 'A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\)', Glasnik matematički, 55(2), str. 195-201. https://doi.org/10.3336/gm.55.2.03
Vancouver
Le M, Soydan G. A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\). Glasnik matematički [Internet]. 2020 [pristupljeno 19.10.2021.];55(2):195-201. https://doi.org/10.3336/gm.55.2.03
IEEE
M. Le i G. Soydan, "A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\)", Glasnik matematički, vol.55, br. 2, str. 195-201, 2020. [Online]. https://doi.org/10.3336/gm.55.2.03

Sažetak
Let \(A\) $B$ be positive integers such that $\min\{A,B\}>1$, $\gcd(A,B) = 1$ and $2|B.$ In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer $n$, if $A >B^3/8$, then the equation $(A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$; if $B>A^3/6$, then it has no solutions $(x,y,z)$ with $y>z>x$. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer $n$, if $B\equiv 2 \pmod{4}$ and $A >B^3/8$, then this equation has only the positive integer solution $(x,y,z)=(1,1,1)$.

Ključne riječi
Ternary purely exponential Diophantine equation

Hrčak ID: 248663

URI
https://hrcak.srce.hr/248663

Posjeta: 224 *