The Lowest-Degree Polynomial with Nonnegative Coefficients Divisible by the n-th Cyclotomic Polynomial

We pose the question of determining the lowest-degree polynomial with nonnegative coefficients divisible by the n-th cyclotomic polynomial Φn(x). We show this polynomial is 1+x+· · ·+x where p is the smallest prime dividing n whenever 2/p > 1/q1+· · ·+1/qk, where q1, . . . , qk are the other (distinct) primes besides p dividing n. Determining the lowestdegree polynomial with nonnegative coefficients divisible by Φn(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.