A196943 Face-diagonal lengths of Euler bricks.

125, 157, 244, 250, 267, 281, 314, 348, 365, 373, 375, 471, 488, 500, 534, 562, 625, 628, 696, 707, 725, 730, 732, 746, 750, 773, 785, 801, 808, 825, 843, 875, 942, 976, 979, 1000, 1037, 1044, 1068, 1095, 1099, 1119, 1124, 1125, 1193, 1220, 1250, 1256, 1335

Offset: 1

Author

Christopher Monckton of Brenchley, Oct 07 2011

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

cf. A195816, A031173, A031174, A031175. Edge lengths of Euler bricks are A195816.

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).