On the lattice constant for |𝑥³+𝑦³+𝑧³|≦1

Spohn, W. G.

doi:10.1090/S0025-5718-1969-0241366-6

On the lattice constant for $\vert x^{3}+y^{3}+z^{3}\vert \leqq 1$
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by W. G. Spohn PDF

Math. Comp. 23 (1969), 141-149 Request permission

Abstract:

There has been no published work on this intractable problem in the Geometry of Numbers since 1946. In 1944 and 1946 L. J. Mordell and H. Davenport gave bounds for the lattice constant in the Journal of the London Mathematical Society. The present attack stems from considering natural lattices with 9 points on the boundary of the region. The points of these lattices which are interior to the region are removed in the most efficient way by applying a convergent linear programming process. Apparently an infinite number of points must be removed in an infinite number of stages. A conjecture is made about the critical lattices for the region and the conjectured value $.948754. \ldots$ is given for the lattice constant.

References

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