On a conjecture of Iizuka

Abstract

TextFor a given odd positive integer n and an odd prime p, we construct an infinite family of quadruples of imaginary quadratic fields Q(d), Q(d+1), Q(d+4) and Q(d+4p2) with d∈Z such that the class number of each of them is divisible by n. Subsequently, we show that there is an infinite family of quintuples of imaginary quadratic fields Q(d), Q(d+1), Q(d+4), Q(d+36) and Q(d+100) with d∈Z whose class numbers are all divisible by n. Our results provide a complete proof of Iizuka's conjecture (in fact a generalization of it) for the case m=1. Our results also affirmatively answer a weaker version of (a generalization of) Iizuka's conjecture for m≥4. VideoFor a video summary of this paper, please visit https://www.youtube.com/watch?v=0wWLcRaxr2E.