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A rational perfect cuboid is a rectangular parallelepiped whose edges andface diagonals are given by rational numbers and whose space diagonal is equalto unity. Its existence is equivalent to the existence of a perfect cuboid withall integer edges and diagonals. Finding such a cuboid or proving itsnon-existence is an old unsolved problem. Recently, based on a symmetryapproach, the equations of a perfect cuboid were transformed to factorequations. The factor equations turned out to be solvable and, being solved,have led to a pair of inverse problems. Our efforts in the present paper aretoward solving these inverse problems. Algebraic parametrizations for theirsolutions using algebraic functions of two rational arguments are found.

On two algebraic parametrizations for rational solutions of the cuboid equations