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I wanted to try some examples with adeles and strong aproximation. Let $\mathfrak{p}_1 = 2+i$ and $\mathfrak{p}_2 = 2-i$ and $\mathfrak{p}_3 = 7$. Can we a single number $x \in \mathbb{Q}(i)$ that's close to all these numbers?

$$ |x - 1|_{\mathfrak{p}_1} < \frac{1}{\sqrt{5}} \text{ and } |x+1|_{\mathfrak{p}_2} < \frac{1}{\sqrt{5}} \text{ and } |x|_{\mathfrak{p}_3} < \frac{1}{7} $$

I think the $p$-adic valuations make sense. Since $5 = (2+i) \times (2-i)$ so that we could have a valuation with $|5| = \frac{1}{5}$ and that $|2 \pm i| = \frac{1}{\sqrt{5}}$.

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