On a Conjecture Concerning the Number of Solutions To

Abstract

Let a, b, c be fixed coprime positive integers with $\min \{ a,b,c \}>1$ . Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$ . We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples, then $c>10^{18}$ , and there are exactly two solutions $(x_1, y_1, z_1)$ , $(x_2, y_2, z_2)$ with $2 \mid x_1$ , $2 \mid y_1$ , $z_1=1$ , $2 \nmid y_2$ , $z_2>1$ , and, taking $a<b$ , we must have $a=2$ , $b \equiv 1 \bmod 12$ , $c \equiv 5\, \mod 12$ , with $(a,b,c)$ satisfying further strong restrictions. These results support a conjecture put forward by Scott and Styer [‘Number of solutions to $a^x + b^y = c^z$ ’, Publ. Math. Debrecen88 (2016), 131–138].