On a conjecture of Iizuka

2021/06/01 by Hoque, Azizul · 1 citation
#11R11 #11R29 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2106.00395

Abstract

For 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.

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