The negative Pell equation and Stevenhagen's conjecture

Pell's equation, $x^2 - Dy^2 = 1$, has been studied since at least the ancient Greeks. It is well-known that Pell's equation is non-trivially soluble

in integers $x$ and $y$ for every squarefree integer $D > 1$. However, the negative Pell equation $x^2 - Dy^2 = -1$ is much more mysterious.

In 1995 Peter Stevenhagen conjectured an asymptotic formula for the number of squarefree integers $D \le X$ for which the negative Pell

equation is soluble. In this talk I will discuss Stevenhagen's conjecture and its recent resolution by Carlo Pagano and myself.