The lowest-degree polynomial with nonnegative coefficients divisible by the \(n\)-th cyclotomic polynomial (Q1953303)

Summary: We pose the question of determining the lowest-degree polynomial with nonnegative coefficients divisible by the \(n\)-th cyclotomic polynomial \(\Phi_n(x)\). We show this polynomial is \(1 + x^{n/p} + \cdots + x^{(p-1)n/p}\) where \(p\) is the smallest prime dividing \(n\) whenever \(2/p > 1/q_1 + \cdots + 1/q_k\), where \(q_1, \ldots, q_k\) are the other (distinct) primes besides \(p\) dividing \(n\). Determining the lowest-degree polynomial with nonnegative coefficients divisible by \(\Phi_n(x)\) remains open in the general case, though we conjecture the existence of values of \(n\) for which this degree is, in fact, less than \((p-1)n/p\).