On Newman polynomials which divide no Littlewood polynomial

Recall that a polynomial P (x) ∈ Z[x] with coefficients 0, 1 and constant term 1 is called a Newman polynomial, whereas a polynomial with coefficients −1, 1 is called a Littlewood polynomial. Is there an algebraic number α which is a root of some Newman polynomial but is not a root of any Littlewood polynomial? In other words (but not equivalently), is there a Newman polynomial which divides no Littlewood polynomial? In this paper, for each Newman polynomial P of degree at most 8, we find a Littlewood polynomial divisible by P . Moreover, it is shown that every trinomial 1+uxa+vxb, where a < b are positive integers and u, v ∈ {−1, 1}, so, in particular, every Newman trinomial 1 + xa + xb, divides some Littlewood polynomial. Nevertheless, we prove that there exist Newman polynomials which divide no Littlewood polynomial, e.g., x9+x6+x2+x+1. This example settles the problem 006:07 posed by the first named author at the 2006 West Coast Number Theory conference. It also shows that the sets of roots of Newman polynomials VN , Littlewood polynomials VL and {−1, 0, 1} polynomials V are distinct in the sense that between them there are only trivial relations VN ⊂ V and VL ⊂ V. Moreover, V = VL ∪ VN . The proofs of several main results (after some preparation) are computational.