Cullen Numbers in Binary Recurrent Sequences

Abstract

In this paper, we prove that the generalized Cullen numbers, C n (s, l) := s n · 2 n + l, where l is an integer and s := (s n ) n≥0 is a sequence of integers satisfying log |s n | < 2 n−1 + O(1), occur only finitely many times in binary recurrent sequences (u n ) n≥0 whose characteristic roots are quadratic units and that satisfy some additional conditions. We also generalize this result in some sense to show that if we take any finite set of prime numbers P and any integer l, and we write u n − l = P Q, where P is a product of powers of the primes from P, and Q is free of primes from P, then there exist two computable constants c 1 and c 2 depending only on the sequence (u n ) n≥0 , the number l, and the given set of primes P, such that for n > c 1 we have log |Q| > |P | c2 .Finally, we find all Cullen numbers, i.e., numbers of the form n · 2 n + 1 and all Woodall numbers, i.e., the numbers of the form n · 2 n − 1, that are either Fibonacci or Pell

Generalized Cullen numbers in linear recurrence sequences

“…Thus, it is enough to consider n > k + 1. Finally, due to [19,Theorem 3], we can suppose that k ≥ 3.…”

“…Now, we combine the estimates (26) and (27) to obtain k < 21761 log m. On the other hand, inequality (19) gives k < 2.1 · 10 6 . Therefore, m < 2.5 · 10 41 .…”

“…Now, we shall prove that there is no solution when k ≥ 159. In this case, (19) implies n < 2.5m + 0.8 < 11.8 · 10 13 k 4 log 2 k + 0.8 < 2 k/2 . Now, we use a key argument due to Bravo and Luca [2, p. 77-78].…”

“…In 2003, Luca and Stȃnicȃ [19] showed, in particular, that Cullen numbers occur only finitely many times in a binary recurrent sequence satisfying some additional conditions. As application, they proved that the largest Fibonacci number in the Cullen sequence is F 4 = 3 = 1 · 2 1 + 1.…”

On Repdigits as Sums of Fibonacci and Tribonacci Numbers

“…c 3 c 2 c 1 (thus it can be said that they are "symmetrical" with respect to an axis of symmetry). The first 19th palindromic numbers are 0, 1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99 and clearly they are a repdigits type. A number n is called repdigit if it has only one repeated digit in its decimal expansion.…”