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Although the original problem asks for integer lattice points in a circle, there is no reason not to consider other shapes, for example conics; indeed Dirichlet's divisor problem is the equivalent problem where the circle is replaced by the rectangular hyperbola. [3]
Dividing a circle into areas – Problem in geometry; Equal incircles theorem – On rays from a point to a line, with equal inscribed circles between adjacent rays; Five circles theorem – Derives a pentagram from five chained circles centered on a common sixth circle; Gauss circle problem – How many integer lattice points there are in a circle
Topics in the first part include the relation between the maximum distance between parallel lines that are not separated by any point of a lattice and the slope of the lines, [5] Pick's theorem relating the area of a lattice polygon to the number of lattice points it contains, [4] and the Gauss circle problem of counting lattice points in a ...
An important example is the Gauss circle problem, which asks for integers points (x y) which satisfy x 2 + y 2 ≤ r 2 . {\displaystyle x^{2}+y^{2}\leq r^{2}.} In geometrical terms, given a circle centered about the origin in the plane with radius r , the problem asks how many integer lattice points lie on or inside the circle.
As of today, this problem remains unsolved. Progress has been slow. Many of the same methods work for this problem and for Gauss's circle problem, another lattice-point counting problem. Section F1 of Unsolved Problems in Number Theory [2] surveys what is known and not known about these problems.
Most of the unsolved problems are related to distribution of Gaussian primes in the plane. Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm ...
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
The spacing between the points determines the noise tolerance of the transmission, while the circumscribing circle diameter determines the transmitter power required. Performance is maximized when the constellation of code points are at the centres of an efficient circle packing. In practice, suboptimal rectangular packings are often used to ...