A NOTE ON THE TERNARY PURELY EXPONENTIAL DIOPHANTINE EQUATION fx+(f+g)y=gz

Abstract

Let f, g be fixed coprime positive integers with min⁡{f,g}>1. Recently, T. Miyazaki and N. Terai [11] conjectured that the equation fx+(f+g)y=gz has no positive integer solutions (x,y,z), except for certain known pairs (f,g). This is a problem that is far from being solved. Let r be an odd positive integer with r>1. In this paper, using Baker’s method with some known results on the generalized Lebesgue-Nagell equations, we prove that if f=2r and one of the following conditions is satisfied, then the above conjecture is true. (i) Either g or f+g has a divisor d with d≡5 or 7 (mod⁡ 8). (ii) f>22493glog⁡g or 167748log⁡g according to g≡1 or 3 (mod⁡ 8).