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Question Number 40974 by Penguin last updated on 30/Jul/18

X = (x_1 , x_2 , ..., x_n ),    X∈P^n      Does there exist a point in N-dimensions,  such that the points and line length (from origin)  are prime?     That is,  Σ_(i=1) ^n x_i ^2 =p^2 ,   p∈P

$${X}\:=\:\left({x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:...,\:{x}_{{n}} \right),\:\:\:\:{X}\in\mathbb{P}^{{n}} \\ $$$$\: \\ $$$$\mathrm{Does}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{N}-\mathrm{dimensions}, \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{points}\:\mathrm{and}\:\mathrm{line}\:\mathrm{length}\:\left(\mathrm{from}\:\mathrm{origin}\right) \\ $$$$\mathrm{are}\:\mathrm{prime}? \\ $$$$\: \\ $$$$\mathrm{That}\:\mathrm{is}, \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{x}_{{i}} ^{\mathrm{2}} ={p}^{\mathrm{2}} ,\:\:\:{p}\in\mathbb{P} \\ $$

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