On some ternary pure exponential diophantine equations with three consecutive positive integers bases

Abstract

By using the lower bound of linear forms in two logarithms of Laurent (Acta Arith. 133(4) (2008) 325–348), we give here a new solution that the ternary pure exponential diophantine equation $$(n+1)^{x}+(n+2)^{y}=n^{z}$$ has no positive integer solutions except for $$(n,x,y,z)=(3,1,1,2)$$ . This proof is very different from Le (J. Yulin Teachers College 28(3) (2007) 1–2), in which he used the classification method of solutions of exponential decomposition form equation. Furthermore, we solved completely another similar ternary pure exponential diophantine equation $$n^{x}+(n+2)^{y}=(n+1)^{z}$$ by using m-adic estimation of linear forms due to Bugeaud (Compos. Math. 132(2) (2002) 137–158).