number theory - The Four Square Theorem and Integral Apollonian Circle Packings, is there any connection? - Mathematics Stack Exchange

I have been studying theta-functions and made an interesting observation which I have a question about QUESTION: Is there a more intuitive, in particular a mostly geometric way, to prove the four

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geometry - Frame challenge: Find the maximum $n$ such that circles of radius $1, rac12, rac13, , rac1n$ can be held immobile by a convex frame. - Mathematics Stack Exchange

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geometry - Intuitively, why aren't all points in a circle covered in an Apollonian gasket? - Mathematics Stack Exchange

PDF) A polytopal generalization of Apollonian packings and Descartes' theorem

geometry - How the recursive structure of Apollonian gaskets can be described in order to be able to reproduce them? - Mathematics Stack Exchange

Apollonian circle packings: number theory - ScienceDirect