An application of the Baker method to a new conjecture on exponential Diophantine equations

Abstract

<abstract><p>Let $ n $ be a positive integer with $ n &gt; 1 $ and let $ a, b $ be fixed coprime positive integers with $ \min\{a, b\} &gt; 2 $. In this paper, using the Baker method, we proved that, for any $ n $, if $ a &gt; \max\{15064b, b^{3/2}\} $, then the equation $ (an)^x+(bn)^y = ((a+b)n)^z $ has no positive integer solutions $ (x, y, z) $ with $ x &gt; z &gt; y $. Further, let $ A, B $ be coprime positive integers with $ \min\{A, B\} &gt; 1 $ and $ 2|B $. Combining the above conclusion with some existing results, we deduced that, for any $ n $, if $ (a, b) = (A^2, B^2), A &gt; \max\{123B, B^{3/2}\} $ and $ B\equiv 2\pmod 4 $, then this equation has only the positive integer solution $ (x, y, z) = (1, 1, 1). $ Thus, we proved that the conjecture proposed by Yuan and Han is true for this case.</p></abstract>