The lattice points of ann-dimensional tetrahedron

Summary

We show that the number of orderedm-tuples of points on the integer lattice, inside or on then-dimensional tetrahedron bounded by the hyperplanesX ₁=0,X ₂=0, ...,X _n=0 andw ₁ X ₁+w ₂ X _n+...+w _n X_n=X, with the property that, for eachj, no more thank such points have non-zerojth ordinate, is asymptotically

$$\left\{ {\prod\limits_{ = 1}^n {\left( {\frac{X}{{w_J }}} \right)} } \right\}^k \times \sum \left( {\begin{array}{*{20}c} n \\ c \\ \end{array} } \right)\frac{1}{{\prod\nolimits_{l = 1}^m {d_t !} }}$$

asX → ∞, where$\left( {\begin{array}{*{20}c} n \\ c \\ \end{array} } \right): = n!/\prod c_I !$, this product and the sum above are taken over all sets$\{ c_l :I \subseteq \{ 1,...,m\} ,|I| = k\}$ of non-negative integers which sum ton, and$d_i : = \Sigma _{Ii \in I^C I}$ for eachi.

As a consequence we deduce estimates for functions that have been used to provide lower bounds for the smallest exception to the first case of Fermat's Last Theorem.