Fibonacci s-Cullen and s-Woodall numbers

Abstract

The m-th Cullen number C m is a number of the form m2 m + 1 and the m-th Woodall number W m has the form m2 m − 1. In 2003, Luca and Stănică proved that the largest Fibonacci number in the Cullen sequence is F 4 = 3 and that F 1 = F 2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given s > 1, the equation F n = ms m ± 1 has only finitely many solutions, and they are effectively computable. In this note, we shall provide the explicit form of the possible solutions.