A195816 Edge lengths of Euler bricks.

44, 85, 88, 117, 132, 140, 160, 170, 176, 187, 195, 220, 231, 234, 240, 252, 255, 264, 275, 280, 308, 320, 340, 351, 352, 374, 390, 396, 420, 425, 429, 440, 462, 468, 480, 484, 495, 504, 510, 528, 550, 560, 561, 572, 585, 595, 616, 640, 660, 680, 693, 700

Offset: 1

Author

Christopher Monckton of Brenchley, Oct 06 2011

Keywords

• nonn

Comments

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

Examples

			For n=1, the edges (a,b,c) are (240,117,44) and the diagonals (d(a,b),d(a,c),d(b,c)) are (267,244,125).
		

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

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Euler brick

Crossrefs

Cf. A031173, A031174, A031175.

Programs

Mathematica

ok[a_] := Catch[Block[{b, c, s}, s = Reduce[a^2 + b^2 == c^2 && b > 0 && c > 0, {b, c}, Integers]; If[s === False, Throw@ False, s = b /. List@ ToRules@ s]; Do[If[ IntegerQ@ Sqrt[s[[i]]^2 + s[[j]]^2], Throw@ True], {i, 2, Length@s}, {j, i - 1}]]; False]; Select[ Range[700], ok] (* Giovanni Resta, Nov 22 2018 *)

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)