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For a given odd positive integer n and an odd prime p, we construct an infinite family of quadruples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and $\mathbb{Q}(\sqrt{d+4p^2})$ with $d\in \mathbb{Z}$ 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 $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$, $\mathbb{Q}(\sqrt{d+36})$ and $\mathbb{Q}(\sqrt{d+100})$ with $d\in \mathbb{Z}$ 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 $m\ge 4$.

Video

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中文翻译：

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关于饭冢的猜想

文本

对于给定的奇正整数n和奇质数p，我们构造了一个无限的四元组虚二次域$\mathbb{问}(\sqrt{d})$,$\mathbb{问}(\sqrt{d+1})$,$\mathbb{问}(\sqrt{d+4})$和$\mathbb{问}(\sqrt{d+4p^2})$和$d\in \mathbb{Z}$使得它们每个的类号都可以被n整除。随后，我们证明了存在一个无限的五元组虚二次域$\mathbb{问}(\sqrt{d})$,$\mathbb{问}(\sqrt{d+1})$,$\mathbb{问}(\sqrt{d+4})$,$\mathbb{问}(\sqrt{d+36})$和$\mathbb{问}(\sqrt{d+100})$和$d\in \mathbb{Z}$其班级编号都可以被n整除。我们的结果为该案例提供了饭冢猜想（实际上是它的推广）的完整证明$米=1$. 我们的结果也肯定地回答了 Iizuka 猜想的较弱版本（概括）$米\ge 4$.

视频

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