Euler Brick

Author

Eric W. Weisstein
February 22, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DiophantineEquations/EulerBrick.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EulerBrick.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Data

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32,27993205,20450307,19208332,28024940.684937},{20409272,8205120,3559215,21996872,20717297,8943825,22282961.858797},{20453760,11170320,4315201,23305200,20904001,11974849,23701335.546977},{20500920,18441753,11214560,27575097,23367800,21583897,29768310.844806},{20641005,3035604,1657840,20863029,20707475,3458804,20928793.862056},{20727252,12627989,11948640,24271075,23924652,17384939,27052820.176559},{20764755,17438512,4582116,27115987,21264309,18030460,27500409.779158},{20808645,762128,742140,20822597,20821875,1063772,20835818.140980},{20911275,13114140,5630768,24683235,21656107,14271868,25317338.689820},{21035700,12427349,7245840,24432349,22248660,14385451,25484149.484795},{21045024,20245893,17512820,29202555,27378676,26769293,34051256.700457},{21066825,15348960,7204648,26065335,22264727,16955752,27042718.825335},{21093504,3469415,700128,21376921,21105120,3539353,21388383.077190},{21114000,15406435,9761148,26137315,23261148,18238373,27900524.111334},{21162960,9537220,5054589,23212700,21758211,10793861,23756647.727508},{21311800,11400480,10055199,24169480,23564801,15201249,26177677.330122},{21318000,2318732,1433835,21443732,21366165,2726243,21491615.223036},{21363888,11940480,8951041,24474288,23163265,14923009,26059775.672301},{21414552,3804320,3169089,21749848,21647775,4951361,21979513.486723},{21415680,20983144,5034183,29982056,21999417,21578585,30401754.562272},{21458580,7896816,4648787,22865484,21956363,9163565,23333271.933564},{21585312,8533575,4959416,23210937,22147720,9870041,23734856.297838},{21589848,17255136,3195665,27638040,21825073,17548561,27822176.942752},{21603840,7067808,2776025,22730592,21781465,7593433,22899478.759812},{21606660,20054272,8662005,29479172,23278275,21845003,30725427.782630},{21612921,18054400,11641680,28161671,24548871,21482320,30473077.080181},{21637200,6349420,1721853,22549580,21705603,6578747,22615223.543667},{21672331,7918020,5395392,23073469,22333835,9581508,23695890.498726},{21760875,19549632,19031540,29252757,28909085,27283468,34898757.956017},{21852675,19674160,6632388,29404285,22836987,20762012,30143001.624652},{21914475,10590860,2990736,24339485,22117611,11005036,24522541.301564},{22096384,16109280,11332575,27345184,24832991,19696095,29600444.998758},{22134000,15167691,12005060,26832309,25180060,19343741,29395480.466818},{22148280,20154400,5112801,29945720,22730751,20792801,30379053.316126},{22244040,5020575,2118688,22803585,22344712,5449313,22901797.477350},{22287265,19752480,17542776,29780575,28363201,26417976,34563443.652460},{22322547,4421604,2217280,22756245,22432397,4946404,22864011.395607},{22331232,6845475,1457300,23356893,22378732,6998875,23402311.294046},{22428225,11188408,200256,25064033,22429119,11190200,25064832.987487},{22432160,8061300,963699,23836660,22452851,8118699,23856132.874341},{22526400,22348444,10091565,31731556,24683565,24521269,33297617.487297},{22713360,19195605,16255468,29738325,27930932,25153757,33891122.933663},{22748973,10066320,2541364,24876627,22890485,10382164,25006101.293037},{22767888,9429084,8018395,24643140,24138587,12377491,25914841.451100},{22815040,16141752,11392689,27947848,25501361,19757265,30180715.208256},{22898304,19594575,1128680,30137679,22926104,19627055,30158806.575351},{22903881,8426880,2979808,24404919,23096905,8938208,24586161.292756},{22923208,13953681,12312960,26836145,26020808,18609519,29526050.572717},{22953315,12179420,6276528,25984475,23795997,13701572,26731774.066837},{23072400,21501753,3967040,31538247,23410960,21864647,31786765.016192}};

The 2001 Monolith is not an Euler Brick

Sqrt[Plus@@(#^2)]&/@Subsets[{1,4,9},{2}]

Sqrt[Plus@@({1,4,9}^2)]

Parametric Solutions

a={3,6,5,9,8,12,15,7,10,20,18,16,21,12,15,24,9,27,30,14,24,20,28,33,40,36,11,39,33,25,16,32,42,48,24,45,21,30,48,18,51,40,36,13,60,39,54,35,57,65,60,28,20,48,40,63,56,60,66,36,15,69,80,45,56,72,22,27,75,44,35};
b={4,8,12,12,15,16,20,24,24,21,24,30,28,35,36,32,40,36,40,48,45,48,45,44,42,48,60,52,56,60,63,60,56,55,70,60,72,72,64,80,68,75,77,84,63,80,72,84,76,72,80,96,99,90,96,84,90,91,88,105,112,92,84,108,105,96,120};
c={5,10,13,15,17,20,25,25,26,29,30,34,35,37,39,40,41,45,50,50,51,52,53,55,58,60,61,65,65,65,65,68,70,73,74,75,75,78,80,82,85,85,85,85,87,89,90,91,95,97,100,100,101,102,104,105,106,109,110,111,113,115,116,117,119,120};

abc=Take[#,Min[Length/@{a,b,c}]]&/@{a,b,c};

#1+#2-#3&@@@(Transpose[abc]^2)

Dickson, Vol. 2, p. 497

PowerExpand[FullSimplify[Sqrt[{#1+#2,#1+#3,#2+#3}&@@({a(4b^2-c^2),b(4a^2-c^2),4a b c}^2)]
]]

PowerExpand[PowerExpand[Simplify[(Sqrt[(b^2) (((-4 (a^2)+(c^2)))^2)+(a^2) (((-4 (b^2)+(c^2)))^2)])/.c->Sqrt[a^2+b^2]]]/.a^2+b^2->c^2]

PowerExpand[Simplify[(Sqrt[(b^2) (((-4 (a^2)+(c^2)))^2)+(a^2) (((-4 (b^2)+(c^2)))^2)])-c^3/.c->Sqrt[a^2+b^2]]]

Union[Reverse/@Sort/@Abs[
Function[{a,b,c},{a(4b^2-c^2),b(4a^2-c^2),4a b c}]@@@Transpose[abc]
]]

Sqrt[{#1+#2,#1+#3,#2+#3}&@@@(%239^2)]

Union[Reverse/@Sort/@Abs[Flatten[Table[With[{a=v^2-u^2,b=2u v,c=u^2+v^2},
{a(4b^2-c^2),b(4a^2-c^2),4a b c}
],{v,10},{u,v+1,10}],1]]]

Sqrt[{#1+#2,#1+#3,#2+#3}&@@@(%226^2)]

Guy p. 174 (I don't understand this)

Flatten [Table[With[{a=4 p q,b=p^2+q^2},{{p,q},{2a b,a^2-b^2,a^2+b^2}}],{p,5},{q,5}],1]//ColumnForm

Search

EulerBrickSearch[n1_,n2_]:=Module[{e={},a,b,c},
	 Do[
			If[EulerBrickQ[a,b,c],
					e=Append[e,{a,b,c}]; Print[e]];
			If[Mod[a,10]==0 && b==a==c,Print["a=",a]],
		 {a,n1,n2},{b,a,n2},{c,b,n2}
		];
		e
]
EulerBrickSearch[n_:100]:=EulerBrickSearch[1,n]

EulerBrickQ[44,117,240]

Exhaustive Search

<<IntegerSequences`

l=ReadList["Phosphine:MathOzTeX:Notebooks:EulerBricks.dat",
	Table[Number ,{7}]];
EulerBrick=Table[Take[l[[i]],3],{i,Length[l]}];

Union[Length/@Join/@EulerBrick]

Do[If[!EulerBrickQ@@EulerBrick[[i]],Print[i]],
	{i,Length[EulerBrick]}]

Table[EulerBrick[[i]],{i,5}]

Table[EulerBrick[[i,1]],{i,100}]

Table[EulerBrick[[i,2]],{i,100}]

Table[EulerBrick[[i,3]],{i,100}]

Diophantine Equation

v5.0

Reduce[dab==Sqrt[a^2+b^2]&&dac==Sqrt[a^2+c^2]&&dbc==Sqrt[b^2+c^2]&&(a|b|c|dab|dac|dbc)∈Integers,{a,b,c}]

Reduce[
a^2+b^2==c^2&&
a^2+d^2==e^2&&
b^2+d^2==f^2&&
b^2+e^2==g^2&&
(a|b|c|d|e|f|g)∈Integers&&
0<a<100,
{a,b,c}]