The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity theory and number the- ory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of a singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, this research project proves the Yau Geometric Con- jecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six- dimensional simplex. This research project paves the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also moves the mathematical community one step closer towards proving the Yau Geometric and Yau Number-Theoretic Conjectures in full generality.

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Our paper begins with a simple but beautiful subject given by Martin Gardner (Dividing ten gold coins and ten silver coins into two containers which look the same outside, then randomly choose one of the containers and take out one coin from the chosen container, Is the probability of getting golden coin becomes higher when the distributive method changes). And we change and consider the problem from some other points of view. 1. Researching the original subject’s probability. 2. Discussing the minimal unit of distributing golden coins. 3. Researching the number of containers. 4. Researching the minimal and maximal probability of getting golden coin which is the function expression of containers’ number. As a result, we will understand the subject more deeply. The essence of the subject tells us: the reason of probability’s changing can not only be interpreted by “Sample Space difference and Sample Point difference”, but also “Classical Probability’s infection to Geometry Probability”. According to this theory, we can model this subject: Gold→A and Silver→not A. Then we use the model in our example, successfully interpreting what J·BERTRAND had found. And we also get a conjecture and prove it. Further, we did some transformation of the original formulation according to the actual situation. This is our paper’s main content.

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