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By R.M. van Luijk
Perfect cuboids are rectangular parallelepipeds of which all sides, face diagonals and full diagonals are integers. They correspond to rational points on an explicit algebraic surface. In this paper we prove that this singular surface is of general type. We also analyse a surface describing cuboids of which all the lengths above are integral, except possibly of one of the face diagonals. This is shown to be a singular K3-surface and we compute its full Néron-Severi group.
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