Generalized Cullen numbers in linear recurrence sequences

Abstract

A Cullen number is a number of the form m2m+1, where m is a positive integer. In 2004, Luca and Stănică proved, among other things, that the largest Fibonacci number in the Cullen sequence is F4=3. Actually, they searched for generalized Cullen numbers among some binary recurrence sequences. In this paper, we will work on higher order recurrence sequences. For a given linear recurrence (Gn)n, under weak assumptions, and a given polynomial T(x)∈Z[x], we shall prove that if Gn=mxm+T(x), then m≪1 and n≪log⁡|x|, where the implied constants depend only on (Gn)n and T(x).