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The Gauss Circle Problem: find the number of integer lattice points inside a circle.

My question is: why was Gauss studying this problem? Was it just math for math's sake, or was this a part of a larger problem he wanted to solve?

I'm guessing there must be a connection to Gaussian integers, but I tried to find the origin on math history books and couldn't find anything.

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Gauss was motivated by arithmetic. He wanted to study the asymptotic behavior of the number of representations of integers as a sum of two squares (of integers).

See, for example, K. Chandrasekharan, Introduction to analytic number theory, Springer, 1968.

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