Topic
Cullen number
About: Cullen number is a research topic. Over the lifetime, 9 publications have been published within this topic receiving 24 citations.
Papers
TL;DR: In this article, it was shown that if a > 1 is any fixed integer, then for a sufficiently large x > 1, the nth Cullen number Cn = n2 n + 1 is a base a pseudoprime only for at most O(x log log x/log x) positive integers nx.
Abstract: We show that if a > 1 is any fixed integer, then for a sufficiently large x > 1, the nth Cullen number Cn = n2 n + 1 is a base a pseudoprime only for at most O(xloglog x/log x) positive integers nx. This complements a result of E. Heppner which asserts that Cn is prime for at most O(x/log x) of positive integers nx. We also prove a similar result concerning the pseudoprimality to base a of the Woodall numbers given by Wn = n2 n 1 for all n � 1.
13 citations
TL;DR: For a given linear recurrence (G n ) n, under weak assumptions, and a given polynomial T ( x ) ∈ Z [ x ], it was shown in this paper that if G n = m x m + T (x ), then m ≪ 1 and n ≪ log | x |, where the implied constants depend only on ( G n n and T( x ).
10 citations
TL;DR: In this article, an algorithm to determine whether or not a given system of congruences is satisfied by Cullen numbers is described. And they use this algorithm to prove that there are infinitely many Cullen numbers which are both Riesel and Sierpinski.
10 citations
1 Jan 2014
TL;DR: This paper searches for Fibonacci numbers belonging to these generalized Cullen and Woodall sequences, defined by Cm, s = ms m +1 and Wm,s =MS m −1, for s > 1.
Abstract: The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m − 1. In 2003, Luca and Stuanicua proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. A generalization of these sequences is defined by Cm,s = ms m +1 and Wm,s = ms m −1, for s > 1. In this paper, we search for Fibonacci numbers belonging to these generalized Cullen and Woodall sequences.
5 citations
1 Jan 2015
TL;DR: This note shall provide the explicit form of the possible solutions of the equation Fn = ms m ± 1, which, for any given s > 1, has only finitely many solutions.
Abstract: The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m 1. In 2003, Luca and Stuanicua proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given s > 1, the equation Fn = 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.
3 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2019 | 1 |
| 2018 | 1 |
| 2015 | 1 |
| 2014 | 1 |
| 2012 | 1 |