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Question:

What is known about the enumeration of all $(a,b,c,d,e,f)\in\mathbb{N}^6_+: \\ \quad\operatorname{GCD}(a,b,c,d,e,f)=1\ \\ \land\ \exists \lbrace x_1,x_2,x_3,x_4\rbrace\subset\mathbb{E}^2:\ \\ \quad \lbrace \|x_j-x_i\| \rbrace_{1\le i\lt j\le 4}=\lbrace a,b,c,d,e,f\rbrace$

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  • $\begingroup$ The term "flat" in the title was chosen instead of "planar" because of the ambigous meaning of the latter when dealing with graph theory and geometry in the same context. $\endgroup$
    – Manfred Weis
    Commented Dec 2, 2023 at 8:44
  • $\begingroup$ What is $\mathbb{E}^2$? Is it $\mathbb{R}^2$? $\endgroup$
    – GH from MO
    Commented Dec 2, 2023 at 17:46
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    $\begingroup$ @GHfromMO its the euclidean plane; I have seen that notation frequently. Using $\mathbb{R}^2$ for the euclidean plane necessitates losing words about the distance metric. $\endgroup$
    – Manfred Weis
    Commented Dec 2, 2023 at 21:17
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    $\begingroup$ Regarding the notation for the Euclidean plane see e.g. Euclidean plane. It may be debatable if $\boldsymbol{E}^2$ and $\mathbb{E}^2$ are both acceptable. $\endgroup$
    – Manfred Weis
    Commented Dec 2, 2023 at 21:37

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