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An infinite family of perfect parallelepipeds


Authors: Benjamin D. Sokolowsky, Amy G. VanHooft, Rachel M. Volkert and Clifford A. Reiter
Journal: Math. Comp. 83 (2014), 2441-2454
MSC (2010): Primary 11D09
DOI: https://doi.org/10.1090/S0025-5718-2013-02791-3
Published electronically: November 18, 2013
MathSciNet review: 3223340
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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

DOI: https://doi.org/10.1090/S0025-5718-2013-02791-3
Received by editor(s): August 9, 2012
Received by editor(s) in revised form: December 27, 2012
Published electronically: November 18, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.