Isolation of the second minimum in the Richaud-Degert family

Prove that, for $m=n^2-2$ with $n\ge5$ odd, the second Euclidean minimum of the corresponding real quadratic field is not isolated.

Background

The paper develops methods for studying isolated and non-isolated Euclidean minima and notes that certain real quadratic families appear to exhibit non-isolated second minima. The conjecture concerns the Richaud-Degert-type family m=n2−2m=n^2-2 and extends the known examples involving non-isolated second minima.

References

we have already expressed the conjecture that in the case $m = n2-2$, $n \ge 5$, $n \equiv 1 \bmod 2$, the second minimum is not isolated

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Section 2, discussion following equation (2.5')