Newman quadrinomials having littlewood multiples

An integer polynomial P is a Newman polynomial if all of its coefficients are in {0, 1} and P(0) = 1. Similarly, an integer polynomial P is a Littlewood polynomial if all of its coefficients are in {−1, 1}. Let P(X) ∈ Z[X] be a monic polynomial without any roots on the complex unit circle |z| = 1 and D ⊂ Z a finite set. In present work an algorithm, that determines whether P(X) divides some polynomial with all the coefficients in D, was used to examine which Newman quadrinomials of degree at most 15 has any Littlewood multiples. We found that for every such quadrinomial, except for possibly eight, there exists a Littlewood multiple, extending previous results obtained by Borwein, Hare, Mossinghoff, Dubickas, Jankauskas, Drungilas and Šiurys. Moreover, Dubickas and Jankauskas proved, that if polynomial P(X) ∈ Z[X] divides a Littlewood polynomial L then deg2 P divides deg L + 1. We asked the following question: given a polynomial P ∈ Z[X] that divides any Littlewood polynomial, does there exist a Littlewood multiple L of P of degree deg L = deg2 P − 1? Note that this is the smallest possible degree of Littlewood multiple. We used the same altgorithm to answer this question for Newman polynomials up to certain degree. The results are as follows. If a Newman polynomial P satisfies the following conditions: 1) polynomial P degree deg P is at most 10, 2) polynomial P divides any Littlewood polynomial, 3) polynomial P is not one of 5 polynomials, specified in 4.3, then P has a Littlewood multiple L of degree deg L = deg2 P −1. Similar statement holds for quadrinomials as well. Moreover, every Newman quadrinomial P of degree at most 14 have a Littlewood multiple L of degree deg L = deg2 P − 1