Existence of a rational 3 × 3 magic square of squares

Determine whether a 3 × 3 magic square of squares with distinct integer entries exists, equivalently whether the corresponding open subvariety of the algebraic surface parameterizing 3 × 3 magic squares of squares has a rational point.

Background

The paper studies the algebraic surface V parameterizing 3 × 3 magic squares of squares. Rational points on V yield such squares after clearing denominators, although additional conditions are required to ensure that the entries are distinct and nonzero. The existence problem has remained unresolved for centuries, despite known examples over various number fields and the existence of larger magic squares of squares.

The authors explicitly distinguish the unresolved rational and integer problem from the geometric results established in the paper. They identify an open subvariety U corresponding to squares with distinct entries and explain that solving the Diophantine problem amounts to proving that U has no rational points, but they do not establish this.