Counting the Number of Integral Points in General n-Dimensional Tetrahedra and Bernoulli Polynomials

Recently there has been tremendous interest in counting the number of integral points in n-dimensional tetrahedra with nonintegral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in n-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous ndimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.