Glasnik matematički, Vol. 55 No. 2, 2020.

Izvorni znanstveni članak

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.org/0000-0002-6321-4132 ; Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey

Puni tekst:

str. 195-201

preuzimanja: 446

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 $(x,y,z)=(1,1,1)$.

248663

23.12.2020.

Posjeta: 997 *