The lattice points of ann-dimensional tetrahedron
Tóm tắt
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 Xn=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.
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