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The lattice points of an \(n\)-dimensional tetrahedron. (English) Zbl 0733.11034

Let w\(=(w_ 1,...,w_ n)\) and x\(=(x_ 1,...,x_ m)\) be vectors of positive real numbers, and let \(k\geq 2\) be an integer. Consider m by n matrices X with non-negative integer entries \(X_{ij}\) such that each column contains at most k non-zero elements. The primary result of the paper is an asymptotic formula for the number of such matrices for \({\mathbf {Xw}}\leq {\mathbf{x}}.\)
Such inequalities, with w consisting of the logarithms of the first few primes, occur in considerations of the first case of Fermat’s Last Theorem, when one uses generalizations of the Wieferich and Mirimanoff congruences, as in Gunderson’s thesis. The most recent such work, by D. Coppersmith [Math. Comput. 54, 895-902 (1990; Zbl 0701.11008)] relates to the inequality \(m^ 2+n^ 2\leq x\) in coprime integers m,n with prime factors at most \(y\ll (\log x)^{1/2}\). The present paper gives, in particular, an asymptotic estimate for the number of such m,n.

MSC:

11P21 Lattice points in specified regions
11D41 Higher degree equations; Fermat’s equation

Citations:

Zbl 0701.11008
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References:

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