Analysis Of Sharp Polynomial Upper Estimate Of Number Of Positive Integral Points In A 5-dimensional Tetrahedra

Recently there has been tremendous interest in counting the numberof integral points of n-dimensional tetrahedra with non-integral vertices due to applica-tions in primality testing and factoring in the number theory and in singularity theory.Let â–³(a₁,...,a_n) be an n-dimensional tetrahedron with non-integral vertices describedby x₁/a1 +···+x_n/a_n ≤1,x₁ ≥0, ..., x_n ≥0 where a₁ ≥···≥a_n are any given positive realnumbers. P(a₁,...,a_n) :=#{(x₁,...,x_n) ∈Zn₊ⁿ |Σ_i=1ⁿ x_i/a_i ≤1}. It was Conjecturedthat n!P(a₁,...,a_n) ≤(a₁ -1)···(a_n -1), where a₁ ≥···≥a_n ≥2. The purposeof this paper is to formulate the conjecture on sharp upper estimate of the number ofintegral points in 5-dimensional tetrahedra with non-integral vertices.