On the factorization of a sum of cyclotomic polynomials

Abstract: In 2000, Charles Nicol conjectured that for $n$ and $m$ integers with $n > m >1$, the sum $\Phi_{n}(x)+\Phi_{m}(x)$ is a product of cyclotomic polynomials and either a constant or an irreducible non-cyclotomic polynomial. Little progress has been made on this conjecture since then. In this talk, I discuss recent joint work with Lilit Martirosyan and London Swan, where, in particular, we show that for primes $p$, $q$ and $\ell$ with $p > q > \ell$ and a non-negative integer $r$, the sum $\Phi_{\ell^{r} p}(x)+\Phi_{\ell^{r} q}(x)$ has this property and determine precisely the cyclotomic polynomials dividing the sum. We also discuss cases of the conjecture in which the number of prime factors of $n$ and $m$ can be arbitrary.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)