There are Salem numbers with trace $-3$ and every degree at least $34$

TL;DR

This paper proves the existence of Salem numbers with trace -3 for all even degrees greater than or equal to 34, combining theoretical and numerical methods, and establishing optimality of the result.

Contribution

It introduces a new existence result for Salem numbers with trace -3 for all sufficiently large even degrees, filling a gap in the known degrees.

Findings

Existence of Salem numbers with trace -3 for all even degrees ≥ 34

No Salem numbers of trace -3 and degree ≤ 30

Expected non-existence at degree 32

Abstract

We prove that there exist Salem numbers with trace $-3$ and every even degree $\geq 34$. Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace $-3$ and degree $\leq 30$, our result is optimal up to possibly the single value $32$, for which it is expected there are no such numbers.