The exponential Diophantine equation $(3am^2-1)^x+(a(a-3)m^2+1)^y=(am)^z$

Authors : Nai-juan Deng, Dan-yao Wu, Ping-zhi Yuan

Pages : 2561-2567

View : 9 | Download : 6

Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :Let $a,\ m$ be positive integers such that $am\not\equiv0\pmod{3}, 2\nmid a$, and $a>3$. We prove that the exponential Diophantine equation $(3am^2-1)^x+(a(a-3)m^2+1)^y=(am)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.
Keywords : Diophantine equation, positive integer solution, Fibonacci number