OFFSET
1,1
COMMENTS
Euler bricks are cuboids all of whose edges and face-diagonals are integers.
It is not known whether any Euler brick with space-diagonals that are integers exists.
825 is the only integer common to the sets of edge lengths and of face-diagonal lengths <= 1000 for Euler bricks.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.
P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.
LINKS
Robin Visser, Table of n, a(n) for n = 1..10000
E. W. Weisstein, MathWorld: Euler brick
FORMULA
Integer edges a > b > c such that integer face-diagonals are d(a,b) = sqrt(a^2 + b^2), d(a,c) = sqrt(a^2 + c^2), d(b,c) = sqrt(b^2 + c^2).
EXAMPLE
For n=1, the edges (a,b,c) are (240,117,44) and the face-diagonals (d(a,b),d(a,c),d(b,c)) are (267,244,125).
Note the pleasing factorizations of the edge-lengths of this least Euler brick: 240 = 15*4^2; 117 = 13*3^2; 44 = 11*2^2.
CROSSREFS
KEYWORD
nonn
AUTHOR
Christopher Monckton of Brenchley, Oct 07 2011
STATUS
approved