A NOTE ON THE NUMBER OF SOLUTIONS OF TERNARY PURELY EXPONENTIAL DIOPHANTINE EQUATIONS

摘要

Abstract Let a, b, c be fixed coprime positive integers with $min {a,b,c}>1$ . We discuss the conjecture that the equation $a^{x}+b^{y}=c^{z}$ has at most one positive integer solution $(x,y,z)$ with $min {x,y,z}>1$ , which is far from solved. For any odd positive integer r with $r>1$ , let $f(r)=(-1)^{(r-1)/2}$ and $2^{g(r)},|, r-(-1)^{(r-1)/2}$ . We prove that if one of the following conditions is satisfied, then the conjecture is true: (i) $c=2$ ; (ii) a, b and c are distinct primes; (iii) $a=2$ and either $f(b)ne f(c)$ , or $f(b)=f(c)$ and $g(b)ne g(c)$ .