# On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

Published on Jan 1, 2021in Acta Arithmetica0.608
· DOI :10.4064/AA200221-16-6
Estimated H-index: 3
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S. Muthukrishnan27
Estimated H-index: 27
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Abstract
Let k \geq 1be a cube-free integer with k \equiv 1 \pmod {9}and \gcd(k, 7\cdot 571)=1 In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields \mathbb{Q}(\sqrt{d}) \mathbb{Q}(\sqrt{d+1})and \mathbb{Q}(\sqrt{d+k^2})with d \in \mathbb{Z}such that the class number of each of them is divisible by 3 This affirmatively answers a weaker version of a conjecture of Iizuka \cite{iizuka-jnt}.
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