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.References
- Richard K. Guy, Unsolved problems in number theory, 3rd ed., Problem Books in Mathematics, Springer-Verlag, New York, 2004. MR 2076335, DOI 10.1007/978-0-387-26677-0
- Jorge F. Sawyer and Clifford A. Reiter, Perfect parallelepipeds exist, Math. Comp. 80 (2011), no. 274, 1037–1040. MR 2772108, DOI 10.1090/S0025-5718-2010-02400-7
- Walter Wyss, Perfect parallelograms, Amer. Math. Monthly 119 (2012), no. 6, 513–515. MR 2928666, DOI 10.4169/amer.math.monthly.119.06.513
- W. Wyss, Rational parallelograms, private communication, 2012.
- J. Sawyer, C.A. Reiter, Auxiliary materials for perfect parallelepipeds exist, http://webbox. lafayette.edu/$\mathtt {\sim }$reiterc/nt/ppe/index.html
- B.D. Sokolowsky, A.G. VanHooft, R.M. Volkert, C.A. Reiter, Auxiliary materials for an infinite family of perfect parallelepipeds, http://webbox.lafayette.edu/$\mathtt {\sim }$reiterc/nt/ppinf/ index.html
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