Hopf's quadratic map and permutation invariant properties of primitive cuboids

Abstract

A primitive cuboid is a rectangular paralleliped with natural edges and inner diagonal that have no common factor and can be identified as primitive solution of the four square Diophantine equation x2+y2+z2=t2. The classical quadratic Hopf map associated to Legesgue's identity is used to study the set of primitive cuboids with odd diagonal. Consider the restriction of the image of the induced integer Hopf map to non-negative integer solutions of the four squares equation, which may include zeros, and are primitive or not, as well as the corresponding relevant subset of its fibre. As a main result, we show that permutations in this subset that belong to a given partition type generate the same number of distinct solutions to the four squares equation. In the special case of a prime diagonal this subset is complete in the sense that it coincides with its fibre. It implies that each partition type in the fibre generate the same number of distinct primitive cuboids. As an application, we use Jacobi's four squares theorem to derive Shanks' theorem stating that there are exactly n primitive cuboids with odd prime diagonal p of the form p=8n+1, p=8n-1, p=8n-5 or p=8n+5. Though more complicated than the original proof, it is remarkable that the Hopf map approach does not use Gauss's formula on the number of primitive three squares representations. Moreover, the alternate proof has the advantage to be constructive and yields an algorithm to generate all primitive cuboids with prime diagonal.