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 distinct 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, we 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.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)