A Newman polynomial has all the coefficients in {0, 1} and constant term 1, whereas a Littlewood polynomial has all coefficients in {−1, 1}. We call P (X) ∈ Z[X] a Borwein polynomial if all its coefficients belong to {−1, 0, 1} and P (0) = 0. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle |z| = 1 has a non-zero multiple in Z[X] with coefficients in a finite set D ⊂ Z, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.

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 = ax n + bx 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).

No abstract

No abstract

Polynomials with all the coefficients in {0, 1} and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in {−1, 1} are called Littlewood polynomials. By exploiting an algorithm developed earlier, we determine the set of Littlewood polynomials of degree at most 12 which divide Newman polynomials. Moreover, we show that every Newman quadrinomial X a + X b + X c + 1, 15 > a > b > c > 0, has a Littlewood multiple of smallest possible degree which can be as large as 32765.