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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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An infinite family of perfect parallelepipeds
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by Benjamin D. Sokolowsky, Amy G. VanHooft, Rachel M. Volkert and Clifford A. Reiter PDF
Math. Comp. 83 (2014), 2441-2454 Request permission

Abstract:

A perfect parallelepiped has edges, face diagonals, and body diagonals all of integer length. We prove the existence of an infinite family of dissimilar perfect parallelepipeds with two nonparallel rectangular faces. We also show that we can obtain perfect parallelepipeds of this form with the angle of the nonrectangular face arbitrarily close to $90^{\circ }$. Finally, we discuss the implications that this family has on the famous open problem concerning the existence of a perfect cuboid. This leads to two conjectures that would imply no perfect cuboid exists.
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Additional Information
  • Benjamin D. Sokolowsky
  • Affiliation: Bucknell University, 211 Trowbridge Lane, Downingtown, Pennsylvania 19335
  • Address at time of publication: 450 Circle Road, West G 204F, Stony Book, New York 11790
  • Email: benjamin.sokolowsky@stonybrook.edu
  • Amy G. VanHooft
  • Affiliation: The College at Brockport, State University of New York, 482 West Avenue, Brockport, New York 14420
  • Email: agvanhooft@rochester.rr.com
  • Rachel M. Volkert
  • Affiliation: University of Northern Iowa, 315 North Guilford Street, Sumner, Iowa 50674
  • Email: volkertr@uni.edu
  • Clifford A. Reiter
  • Affiliation: Lafayette College, Department of Mathematics, Easton, Pennsylvania 18042
  • Email: reiterc@lafayette.edu
  • Received by editor(s): August 9, 2012
  • Received by editor(s) in revised form: December 27, 2012
  • Published electronically: November 18, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 2441-2454
  • MSC (2010): Primary 11D09
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02791-3
  • MathSciNet review: 3223340