If the space diagonal is also an integer, the Euler brick is called a perfect cuboid, although no examples
of perfect cuboids are currently known.
The smallest Euler brick has sides and face polyhedron
diagonals ,
, and , and was discovered by Halcke (1719; Dickson 2005,
pp. 497-500). Kraitchik gave 257 cuboids with the odd
edge less than 1 million (Guy 1994, p. 174). F. Helenius has compiled a
list of the 5003 smallest (measured by the longest edge) Euler bricks. The first
few are (240, 117, 44), (275, 252, 240), (693, 480, 140), (720, 132, 85), (792, 231,
160), ... (OEIS A031173, A031174,
and A031175).
Interest in this problem was high during the 18th century, and Saunderson (1740) found a parametric solution always giving Euler bricks (but not giving all possible
Euler bricks), while in 1770 and 1772, Euler found at least two parametric solutions.
Saunderson's solution lets
be a Pythagorean triple, then
Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover,
2005.Guy, R. K. "Is There a Perfect Cuboid? Four Squares Whose
Sums in Pairs Are Square. Four Squares Whose Differences are Square." §D18
in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 173-181,
1994.Halcke, P. Deliciae Mathematicae; oder, Mathematisches sinnen-confect.
Hamburg, Germany: N. Sauer, p. 265, 1719.Leech, J. "The
Rational Cuboid Revisited." Amer. Math. Monthly84, 518-533, 1977.
Erratum in Amer. Math. Monthly85, 472, 1978.Peterson,
I. "MathTrek: Euler Bricks and Perfect Polyhedra." Oct. 23, 1999.
http://www.sciencenews.org/sn_arc99/10_23_99/mathland.htm.Sloane,
N. J. A. Sequences A031173, A031174,
and A031175 in "The On-Line Encyclopedia
of Integer Sequences."Rathbun, R. L. "Integer Cuboid
Search Update." 8 Jan 2001. http://listserv.nodak.edu/scripts/wa.exe?A2=ind0101&L=NMBRTHRY&P=1272.Saunderson,
N. The Elements of Algebra in 10 Books, Vol. 2. Cambridge, England: University
Press, pp. 429-431, 1740.Spohn, W. G. "On the Integral
Cuboid." Amer. Math. Monthly79, 57-59, 1972.Spohn,
W. G. "On the Derived Cuboid." Canad. Math. Bull.17,
575-577, 1974.Wells, D. G. The
Penguin Dictionary of Curious and Interesting Numbers. London: Penguin, p. 127,
1986.