Let $d \equiv 5 \bmod 8$ be a square-free positive integer and consider the fundamental unit $u_d$ of the real quadratic field $K = \mathbb{Q}(\sqrt{d})$. Since 2 is inert in $K$, there are three possible residue classes of $u_d$ modulo (the prime above) 2. All other things being equal, one expects each of the three residue classes to occur equally often. In particular, one expects $u_d \equiv 1$ one third of the time; we call such $d$’s Eisenstein discriminants. Stevenhagen showed in 1990s that Eisenstein discriminants $d$ are related to cubic number fields of discriminant $4d$.

In this talk, I will explore this relationship and in particular compare the counting functions of Eisenstein discriminants and of cubic fields of discriminant $4d$. Some results can be proved, but tantalising mysteries remain.