A binary linear recurrence sequence of composite numbers

作者：Dubickas Arturas^*; Novikas Aivaras; Siurys Jonas

来源：Journal of Number Theory, 2010, 130(8): 1737-1749.

DOI：10.1016/j.jnt.2010.03.015

• Binary recurrence
• Composite number
• Covering systems
• Divisibility sequence

Let (a, b) is an element of Z(2), where b not equal 0 and (a, b) not equal (+/- 2, -1). We prove that then there exist two positive relatively prime composite integers x(1), x(2) such that the sequence given by x(n+1) = ax(n) + bx(n) = 2,3, ... consists of composite terms only, i.e., vertical bar xn vertical bar is a composite integer for each n is an element of N. In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs (a, +/- 1), computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where (a, b) = (1,1).

• 出版日期2010-8