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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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For finding a formula for the number of classes of binary quadratic forms with given discriminant, counting the number of lattice points inside ellipses and hyperbolas is a natural approach. Gauss must have known about this prior to 1801, as he states some results he had obtained using analytic means in his Disquisitiones.

Apparently it is not very well known that Legendre studied the circle problem in his Essai sur la theorie des nombres (see $\S$ 319) published in 1798, also in connection with binary quadratic forms, but on a more elementary level: His aim was provind that a form $Ax^2 + Bxy + Cy^2$ represents a small number $c$ (less than the determinant $D$ of the form) coprime to $D$.

The connection between the circle problem and Gaussian integers is described in a masterly manner in the book "From Fermat to Minkowski" by Scharlau and Opolka.

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