Perfect Cuboid

Author

Eric W. Weisstein
April 29, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

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Square roots

eqn=Sqrt[Total/@(Subsets[{a,b,c},{2,3}]^2)]

Integer space diagonal and two out of three face diagonals:

eqn/.{a->672,b->153,c->104}

Integer space and face diagonals with only a single nonintegral polyhedron edge:

eqn/.{a->18720,b->(Sqrt[211773121]),c->7800}

Quadratic

With[{max=(10^2)},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}<max,{𝒜,ℬ,𝒞,𝒟,ℰ,ℱ,𝒢},PositiveIntegers]]//Timing

With[{max=(10^3)},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}<max,{𝒜,ℬ,𝒞,𝒟,ℰ,ℱ,𝒢},PositiveIntegers]]//Timing

With[{max=(10^6)},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}<max,{𝒜,ℬ,𝒞,𝒟,ℰ,ℱ,𝒢},PositiveIntegers]]//Timing