Consecutive-shift quadratic-field class-number divisibility conjecture

The conjecture predicts infinitely many consecutive tuples of real quadratic fields whose class numbers share a prescribed prime divisor. The newly reported result covers two fields, but not arbitrarily long tuples.

September 2026 two-field theorem

Yoshichika Iizuka claims infinitely many pairs $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D+m})$ with both class numbers divisible by every prescribed integer $N\ge 2$ and fixed $m\ge 1$. For odd $N$, the class groups also contain elements of order $N$; the remaining case $m\ge 2$ of the original consecutive-tuple conjecture is not settled. This advance is reported by the paper but remains unverified here.

Current status (as of September 2026): The two-field case is claimed proved for arbitrary $N$ and fixed shift $m$; tuples of three or more consecutive fields remain open.