Combinatorics and Graph Theory


The Exact Solution of Gauss’s Problem on the Number of Integer "Points" in a Circular and Spherical "Layers"

Authors: Arsen A. Movsesyan

In the article, the Gauss’s problem on the number of integer points for a circle and a ball in the framework of an integer lattice is reformulated in an equivalent way and reduces to solving two combinatorial tasks for a circular and spherical "layer" in the framework of Quantum Discrete Space. These tasks are solved using trigonometric functions defined on a set of integers whose range of values is also integers, and other new mathematical tools. It comes not about evaluative solutions, but about exact solutions, which, if necessary, can be transferred to a circle and a ball. In doing so not only specific formulas for determine the exact number of solutions are presented, but also the formulas for enumerating the corresponding pairs and triples of integers. The importance of obtained solutions lies in the fact that they determine the analytical likenesses of not only the circumference and the sphere in the Quantum Discrete Space, but also point to the possibility of constructing of the likenesses of ellipse, cone, hyperboloid and other figures.

Comments: 23 Pages.

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Submission history

[v1] 2017-11-27 01:02:26
[v2] 2018-01-28 13:19:20

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