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

1,1

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

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

Sequence in context: A126895 A202240 A069656 * A307386 A307515 A038513

Adjacent sequences: A196940 A196941 A196942 * A196944 A196945 A196946

Keyword

nonn

Author

Christopher Monckton of Brenchley, Oct 07 2011

Status

approved