Computing The Number of Integral Points in4-dimensional Ball Using Tutte Polynomial

Abstract

In recent years, the uses of high dimensional appear in a large and a lot of applications appearwithin it. So, we study these applications and take one of them that play a central role in the factoring of prime number which is an application especially in cryptography. Our main purpose is to introduce another procedure which make the operation of computing the factoringof N = p.q as more easy as the direct computation fast, therefore, an approachis working on for finding the number of integral points(lattice points) make benefit from the concept of theTuttepolynomial and its application on integral points of a polytope. Polytopes whichare taken are the Platonic solid, and a map is making between a ball and a polytope in four dimensions, then discusses the relation between the numbers of integral points ofthem from dimensionone to n dimension. We found a relation between the radiuses of the ball, the edge of the cubewhich is one of the Platonic solid and the dimension together with Pascal triangle,the rhombic dodecahedron, octahedron, and icosahedrons are also taken.

Keywords

• polytope
• lattice point
• tutte polynomial