A196943 - OEIS

%I #17 Jan 01 2024 23:53:17

%S 125,157,244,250,267,281,314,348,365,373,375,471,488,500,534,562,625,

%T 628,696,707,725,730,732,746,750,773,785,801,808,825,843,875,942,976,

%U 979,1000,1037,1044,1068,1095,1099,1119,1124,1125,1193,1220,1250,1256,1335

%N Face-diagonal lengths of Euler bricks.

%C Euler bricks are cuboids all of whose edges and face-diagonals are integers.

%C It is not known whether any Euler brick with space-diagonals that are integers exists.

%C 825 is the only integer common to the sets of edge lengths and of face-diagonal lengths <= 1000 for Euler bricks.

%D L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.

%D P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.

%H Robin Visser, <a href="/A196943/b196943.txt">Table of n, a(n) for n = 1..10000</a>

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/EulerBrick.html">MathWorld: Euler brick</a>

%F 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).

%e 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).

%e 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.

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

%K nonn

%O 1,1

%A _Christopher Monckton of Brenchley_, Oct 07 2011