Does there exist a rectangular cuboid whose edges and face diagonals are all integer lengths?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
Cuboids whose edges and face diagonals are all integer lengths are called Euler bricks .
In 1740, Saunderson found that for any Pythagorean triple ( u , v , w ) , the sides a = u ∣ 4 v 2 − w 2 ∣ , b = v ∣ 4 u 2 − w 2 ∣ , and c = 4 u v w produce diagonals d = w 3 , e = u ( 4 v 2 + w 2 ) , and f = v ( 4 u 2 + w 2 ) , which are all integers.
With these formulas, if g cd ( u , v , w ) = 1 , then g cd ( a , b , c ) = 1 . Since there are infinitely many relatively prime Pythagorean triples, there are infinitely many rectangular cuboids with edges and face diagonals such that g cd ( a , b , c ) = 1 .