On certain multiples of Littlewood and Newman polynomials

TL;DR

This paper investigates the divisibility relationships between Littlewood and Newman polynomials, identifying specific Littlewood polynomials that divide Newman polynomials up to degree 12 and analyzing minimal degrees of multiples for certain Newman quadrinomials.

Contribution

It determines the set of Littlewood polynomials dividing Newman polynomials of degree at most 12 and establishes bounds on the degrees of Littlewood multiples for specific Newman quadrinomials.

Findings

Identified Littlewood polynomials dividing Newman polynomials up to degree 12.

Showed minimal degree of Littlewood multiples for certain Newman quadrinomials can be as large as 32765.

Developed an algorithm to facilitate these divisibility determinations.

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 $32765$.