Infinitude of Wall–Sun–Sun primes: infinite_isWallSunSunPrime_of_disc_eq

Let $K$ be a real quadratic field of discriminant $D$ and let $\varepsilon$ be a fundamental unit of $K$. Following [EJ10, Remark 2.2.8], an odd prime $p \nmid D$ is a Wall–Sun–Sun prime for $K$ if, in $\mathcal{O}_K$, $\varepsilon^{p-1} \equiv 1 \pmod{p^2}$ when $\left(\tfrac{D}{p}\right) = 1$, and $\varepsilon^{2p+2} \equiv 1 \pmod{p^2}$ when $\left(\tfrac{D}{p}\right) = -1$. Both exponents are even, so the condition does not depend on the choice of $\varepsilon$, and it is equivalent to asking the same congruence for every unit of $K$. For $K = \mathbb{Q}(\sqrt{5})$ and $\varepsilon = \frac{1 + \sqrt{5}}{2}$ these are the classical Wall–Sun–Sun primes other than $2$ and $5$ [EJ10, Proposition 2.2.6].

It is conjectured that for every fundamental discriminant $D \neq 1$ there are infinitely many Wall–Sun–Sun primes with discriminant $D$ (Wikipedia; [EJ10, §4.1] gives the heuristic for $\mathbb{Q}(\sqrt{5})$). It is stated here for $D > 0$ only. Wikipedia's sentence also covers $D < 0$, but [EJ10] gives the definition above only for real quadratic fields, and its literal extension to imaginary quadratic fields is degenerate: there every unit is a root of unity of order dividing $4$ or $6$, and that order divides the relevant exponent $p - 1$ or $2p + 2$ for every odd prime $p \nmid D$.