Perfect parallelepipeds exist
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- by Jorge F. Sawyer and Clifford A. Reiter PDF
- Math. Comp. 80 (2011), 1037-1040 Request permission
Abstract:
There are parallelepipeds with edge lengths, face diagonal lengths and body diagonal lengths that are all positive integers. In particular, there is a parallelepiped with edge lengths $271$, $106$, $103$, minor face diagonal lengths $101$, $266$, $255$, major face diagonal lengths $183$, $312$, $323$, and body diagonal lengths $374$, $300$, $278$, $272$. Focused brute force searches give dozens of primitive perfect parallelepipeds. Examples include parallellepipeds with up to two rectangular faces.References
- Daniel S. D’Argenio and Clifford A. Reiter, Families of nearly perfect parallelepipeds, JP J. Algebra Number Theory Appl. 9 (2007), no. 1, 105–111. MR 2407809
- 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
- John Leech, The rational cuboid revisited, Amer. Math. Monthly 84 (1977), no. 7, 518–533. MR 447106, DOI 10.2307/2320014
- Jsoftware, http://www.jsoftware.com.
- Randall Rathbun, Private Communication.
- Clifford A. Reiter and Jordan O. Tirrell, Pursuing the perfect parallelepiped, JP J. Algebra Number Theory Appl. 6 (2006), no. 2, 279–294. MR 2283937
- J. Sawyer, C. Reiter, Auxiliary materials for perfect parallelepipeds exist, http://www.lafayette.edu/$\sim$reiterc/nt/ppe/index.html.
- Jordan O. Tirrell and Clifford A. Reiter, Matrix generation of the Diophantine solutions to sums of $3\le n\le 9$ squares that are square, JP J. Algebra Number Theory Appl. 8 (2007), no. 1, 69–80. MR 2370192
Additional Information
- Jorge F. Sawyer
- Affiliation: Box 8681 Farinon Center, Lafayette College, Easton, Pennsylvania 18042
- Email: sawyerj@lafayette.edu
- Clifford A. Reiter
- Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042
- Email: reiterc@lafayette.edu
- Received by editor(s): November 16, 2009
- Received by editor(s) in revised form: December 3, 2009
- Published electronically: August 17, 2010
- Additional Notes: The support of a Lafayette EXCEL grant is appreciated
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 80 (2011), 1037-1040
- MSC (2010): Primary 11D09
- DOI: https://doi.org/10.1090/S0025-5718-2010-02400-7
- MathSciNet review: 2772108