A note on the exponential Diophantine equation \(a^x+b^y=c^z\)
Let \(a\), \(b\), \(c\) be fixed coprime positive rational integers all \({}>1\). In 1933, Mahler proved that the equation \[ a^x+b^y=c^z,\qquad x, y, z \in \mathbb Z, \] has only finitely manysolutions. The ``Terai-Jeśmanowisz'' conjecture states that the above equation has at most a solution \((x,y,z)\) with \(\min\{x,y,z\}>1\). In the present paper, the author proves: Theorem. If \(a\), \(b\), \(c\) are positive rational integers with \(b\equiv 3 \pmod 4\), \(a\equiv 1 \pmod {b^{2l}}\), \(a^2+b^{2l-1}=c\) and \(c\) is odd, where \(l\) is a positive integer, then the above equation has only the solution \((2,2l-1,1)\). This improves on a previous result of Terai. The main tool is a deep result of \textit{Y. Bilu, G. Hanrot} and \textit{P. M. Voutier} [J. Reine Angew. Math. 539, 75--122 (2001; Zbl 0995.11010)] about the existence of primitive divisors of Lucas sequences.
scientific article; zbMATH DE number 2109876