tailieunhanh - Báo cáo toán học: "On Rowland’s sequence Fernando Chamizo, Dulcinea Raboso and Seraf"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On Rowland’s sequence Fernando Chamizo, Dulcinea Raboso and Seraf´. | On Rowland s sequence Fernando Chamizo Dulcinea Raboso and Serafin Ruiz-Cabello Department of Mathematics Universidad Autonoma de Madrid 28049 Madrid. Spain seraf Submitted Apr 30 2011 Accepted May 20 2011 Published May 29 2011 Mathematics Subject Classification 11A41 11B37 Thank you Professor Zeilberger for the unforgettable Experimental Mathematics Seminars 2009-2010 Abstract E. S. Rowland proved that ak ak-1 gcd k ak-1 a1 7 implies that ak ak-1 is always 1 or prime. Conjecturally this property also holds for any a1 3 from a certain k onwards. We state some properties of this sequence for arbitrary values of a1. Namely we prove that some specific sequences contain infinitely many primes and we characterize the possible finite subsequences of primes. 1 Introduction In 4 E. S. Rowland introduced the recursively defined sequence ak ak-1 gcd k afc-1 ai 7. 1 He proved the following suprising result Theorem Rowland 4 Let P be the set of prime numbers and P1 P u 1 . Then ak ak-1 G P1 for every k 1. The first and the third authors are supported by the grant MTM2008-03880 of the Ministerio de Ciencia e Innovacion. The second author is also a member of ICMAT. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P10 1 Unfortunately it is not clear whether the proof applies to all possible values of a1. Note that a1 2A and a1 2A 1 give the same a2 so we can restrict ourselves to odd initial conditions. It is easy to check that a1 1 and a1 3 lead to the sequences ak k and ak k 2 respectively. Hence in this paper we only consider the sequences ak ak-1 gcd k ak-1 with a1 odd and greater than 3. 2 Conjecture For any sequence of the form 2 there exists a positive integer N such that ak ak-1 G P1 for every k N. Actually in 4 this conjecture is stated for starting values of the form ako A. We consider the former statement more natural although less general and as we shall see there are some differences between the .