On the Diophantine equation 1+2a+xb=yn1+2a+xb=yn

Recently, mixed polynomial–exponential equations similar to the one in the title have been considered by many authors. In these results a certain non-coprimality condition plays an important role. In this paper we completely solve the title equation for odd positive integers x   with x<50x<50. Since we avoid the mentioned non-coprimality condition, this can be considered as a partial completion of the above mentioned results. It seems that the deep effective tools (such as Baker's method) alone are not capable to handle the problem. We combine local arguments and Baker's method to prove our results.