A binary linear recurrence sequence of composite numbers

Abstract

Let ( a , b ) ∈ Z 2 , where b ≠ 0 and ( a , b ) ≠ ( ± 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 = a x n + b x n − 1 , n = 2 , 3 , … , consists of composite terms only, i.e., | x n | is a composite integer for each n ∈ 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 ) .