A note on the exponential Diophantine equation \(a^x+db^y=c^z\) (Q5946900)

The author gives an elementary proof of the following result.NEWLINENEWLINENEWLINETheorem. If positive rational integers \(a\), \(b\), \(c\) and \(d\) satisfy NEWLINE\[NEWLINE a^2+db^2=c^2, \quad a \equiv 3 \pmod 8, \quad 4\|b, \quad \left({b\over a}\right)=-1 NEWLINE\]NEWLINE and NEWLINE\[NEWLINE a=db_2^2-b_1^2, \quad b=2b_1b_2,\quad c=db_2^2+b_1^2, NEWLINE\]NEWLINE where \(b_1\) and \(b_2\) are coprime positive integers with \(b_1>1\), \(b_1\equiv 1\pmod 4\), \(2\|b_2\) then NEWLINE\[NEWLINE a^x+db^y=c^z \Rightarrow (x,y,z)=(2,2,2). NEWLINE\]NEWLINE A similar result was obtained by \textit{N. Terai} and \textit{K. Takakuwa} [Tokyo J. Math. 22, 75--82 (1999; Zbl 0940.11018)], but they used linear forms in logarithms.