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Wolfram Alpha will provide integer solutions to arbitrary circle equations. I'm trying to understand how it's able to calculate them, but despite a fair bit of digging I haven't found any discussion of how to get either the number of, or which, integer solutions to a given circle. Plenty of discussion of lattice points inside a circle, related to the Gauss circle problem, and some discussion of circles centered on the origin, but nothing for the general case.

Wolfram Alpha can quickly determine there are $12$ integer solutions to the circle $x^2-10 (x+y)+y^2+50 = 50$ - how?

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    Someone should read through this paper, and figure out the exact ${\cal O}(\cdot)$ time complexity: *Felipe Cucker Pascal Koiran and Steve Smale, A Polynomial Time Algorithm for Diophantine Equations in One Variable* [PDF file](http://lara.inist.fr/bitstream/handle/2332/690/RR1997-45.pdf;jsessionid=C217F2640B9B9606C52C13171D9C13C6?sequence=1)2012-03-25
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    How did you get Wolfram alpha to provide integer solutions?2013-10-29
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    @LarsH I just dropped in the equation and it provided them, see the link in my question.2013-10-29

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