Second minima for unresolved real quadratic fields

Determine the second Euclidean minima $M_2(K)$ for the real quadratic fields with $m=19$, $m=22$, and $m=41$ identified in the paper.

Background

The paper tabulates known first and second Euclidean minima for real quadratic fields up to the stated discriminant range. It explicitly identifies the smallest parameters for which the second minimum remained undetermined and records only lower bounds in those cases.

References

The smallest $m$ for which $M_2(K)$ is still unknown are now $m = 19$ and $m=22$ for $m = 2,3 \bmod 4$, as well as $m=41$ for $m=1 \bmod 4$; in these cases lower bounds for $M_2(K)$ are given in the table.

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Section 2, discussion immediately preceding Tables of Euclidean minima