How many integer lattice points there are in a circle
A circle of radius 5 centered at the origin has area 25π, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. The Gauss circle problem concerns bounding this error more generally, as a function of the radius of the circle.
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius . This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area.
The first progress on a solution was made by Carl Friedrich Gauss, hence its name.
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In mathematics, the Gausscircleproblem is the problem of determining how many integer lattice points there are in a circle centered at the origin and...
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Johann Carl Friedrich Gauss (German: Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs] ; Latin: Carolus Fridericus Gauss; 30 April 1777 – 23 February 1855) was a German mathematician...
today, this problem remains unsolved. Progress has been slow. Many of the same methods work for this problem and for Gauss'scircleproblem, another lattice-point...
pair? The Gausscircleproblem: how far can the number of integer points in a circle centered at the origin be from the area of the circle? Grand Riemann...
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the table below: Integer partition Jacobi's four-square theorem Gausscircleproblem P. T. Bateman (1951). "On the Representation of a Number as the Sum...
to some measure of "size" or height. An important example is the Gausscircleproblem, which asks for integers points (x y) which satisfy x 2 + y 2 ≤ r...
the article on the Gausscircleproblem, one can compute this by approximating the number of lattice points inside of a quarter circle centered at the origin...
According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that such functions did not exist. It was conjectured...
us mention two classical examples, Dirichlet’s divisor problem and the Gauss’ circleproblem. The former estimates the size of d(n), the number of positive...
theory. Most of the unsolved problems are related to distribution of Gaussian primes in the plane. Gauss'scircleproblem does not deal with the Gaussian...
number of lattice points it contains, and the Gausscircleproblem of counting lattice points in a circle centered at the origin of the plane. The second...
Polygon Carlyle circle Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969). Gauss, Carl Friedrich...
area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter,...
to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved...
problem of trying to find the best visual fit of circle to a set of 2D data points. The method elegantly transforms the ordinarily non-linear problem...
Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga...
mathematical fields of differential geometry and geometric analysis, the Gauss curvature flow is a geometric flow for oriented hypersurfaces of Riemannian...
its area. The Gausscircleproblem concerns the error that would be obtained by using a dot planimeter to estimate the area of a circle. As its name suggests...
map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces...
proof of the conjecture, and the next step was taken by Carl Friedrich Gauss (1831), who proved that the Kepler conjecture is true if the spheres have...
integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory...
theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with...
In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst...
a new geometry. Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "non-Euclidean...
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