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A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that $m\le 42$. Our proof is relatively simple compared to the proofs of similar results in positive integers.

Let $k$ be a number field, $O_k$ its ring of integers, $\mathrm{\text{Cl}}(k)$ its classgroup and $h$ the class number of $k$. Let $\mathrm{\Gamma }$ be a finite group. Let ${ℳ}$ be a maximal $O_k$-order in the semi-simple algebra $k[\mathrm{\Gamma }]$ containing $O_k[\mathrm{\Gamma }]$, and $\mathrm{\text{Cl}}({ℳ})$ its locally free classgroup. Let $A=\mathbb{Z}\cup \{\frac{1}{m},m\in \mathbb{Z}\setminus \{0,1,-1\}\}$ and $n\in A$. We define the set ${ℛ}({{𝒟}}^n,{ℳ})$ of Galois module classes realizable by the $n$th power of the different to be the set of classes $c\in \mathrm{\text{Cl}}({ℳ})$ such that there exists a Galois extension $N/k$ with Galois group isomorphic to $\mathrm{\Gamma }$ ($\mathrm{\Gamma }$-extension), which is tamely ramified, and for which the class of ${ℳ}{\otimes }_{O_k[\mathrm{\Gamma }]}{{𝒟}}_{N/k}^n$ is equal to $c$, where we clarify that if $n=\frac{1}{m}$, where $m\in \mathbb{Z}\setminus \{0,1,-1\}$, ${{𝒟}}_{N/k}^n$ is the $\left|m\right|$th root of the inverse different ${{𝒟}}_{N/k}^{-1}$ (respectively, the different ${{𝒟}}_{N/k}$) if $m<0$ (respectively, $m>0$) when it exists. Let $l$ be a prime number and $\xi$ be a primitive $l$th root of unity. In this article, we suppose that $\mathrm{\Gamma }$ is cyclic of order $l$ and $k/\mathbb{Q}$ and $\mathbb{Q}(\xi )/\mathbb{Q}$ are linearly disjoint. We prove, sometimes under an assumption on $h$, that ${ℛ}({{𝒟}}^n,{ℳ})$ is a subgroup of $\mathrm{\text{Cl}}({ℳ})$, by an explicit description using a Stickelberger ideal. In addition, for each $n\in A$, we determine the set of the Steinitz classes of ${{𝒟}}_{N/k}^n$, $N/k$ runs through the tame $\mathrm{\Gamma }$-extensions of $k$, and prove that it is a subgroup of $\mathrm{\text{Cl}}(k)$, also sometimes under an hypothesis on $h$.