Perfect parallelepipeds exist

Sawyer, Jorge; Reiter, Clifford

doi:10.1090/S0025-5718-2010-02400-7

Perfect parallelepipeds exist
HTML articles powered by AMS MathViewer

by Jorge F. Sawyer and Clifford A. Reiter;

Math. Comp. 80 (2011), 1037-1040

DOI: https://doi.org/10.1090/S0025-5718-2010-02400-7

Published electronically: August 17, 2010

PDF | 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