Generalized Cullen numbers in linear recurrence sequences

Abstract

A Cullen number is a number of the form m2 m + 1, where m is a positive integer. In 2004, Luca and Stȃnicȃ proved, among other things, that the largest Fibonacci number in the Cullen sequence is F 4 = 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 (G n ) n , under weak assumptions, and a given polynomial T (x) ∈ Z[x], we shall prove that if G n = mx m + T (x), then m ≪ log log |x| log 2 (log log |x|) and n ≪ log |x| log log |x| log 2 (log log |x|), where the implied constant depends only on (G n ) n and T (x).

The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits

“…The last equation holds for only finitely many values of n. In the case where the inequality b + 1 < rad(b(b + 1)) 2 does not hold, the abc conjecture with ǫ = 1 says that we only have examples for finitely many values of b. Therefore, there are only finitely many n such that C s,n is a repunit of length three.…”

“…As the abc conjecture, with ǫ = 1, implies that there are at most finitely many pairs (x, m) such that x m > rad(x(x m − 1)) 2 , there are finitely many s-Cullen numbers for any fixed s that are also repunits of length greater than three.…”

“…This result was generalized to s-Cullen numbers (for fixed s) by Marques [7]. Bilu, Marques, and Togbé [2] (among other things) generalized to the intersection of s-Cullen numbers with other recurrence sequences.…”

Cullen numbers in sums of terms of recurrence sequence

“…Recently, Bilu et. al., [3] studied the occurrence of generalized Cullen numbers in a fixed linear recurrence sequences. In particular, they proved that there are finitely many solutions in integers (n, m, x) of the Diophantine equation…”

“…However, there is an error in their proof [3,Theorem 1]. For instance, consider the linear recurrence sequence of order three defined by the recurrence relation…”