Iizuka’s conjecture on simultaneous p-divisibility for successive quadratic fields

Establish that for any prime p and any positive integer m, there exists an infinite family of m+1 consecutive quadratic fields Q(√d), Q(√(d+1)), …, Q(√(d+m))—either all real or all imaginary—such that the class number of each field is divisible by p.

Background

Iizuka proved the existence of infinitely many pairs of imaginary quadratic fields Q(√d) and Q(√(d+1)) with class numbers divisible by 3 and then formulated a broader conjecture for arbitrary primes p and lengths m of successive fields. Subsequent work has resolved several special cases (notably m=1 for all primes), but the general conjecture remains a guiding open problem, motivating constructions of tuples with prescribed divisibility, including the quadruples of real quadratic fields developed in this paper.

References

This helped him to frame the following conjecture. For any prime number p and any positive integer m, there is an infinite family of m + 1 successive real (or imaginary) quadratic fields, \mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}), \cdots, \mathbb{Q}(\sqrt{d+m}) with d\in \mathbb{Z} whose class numbers are all divisible by p.

— On the simultaneous $3$-divisibility of class numbers of quadruples of real quadratic fields  (2512.11346 - Banerjee et al., 12 Dec 2025) in Conjecture (Section 1, Introduction)