Third positive primitive solution to Fermat’s square hypotenuse and square sum problem

Construct a third positive primitive solution in integers to the simultaneous Diophantine system x^2 + y^2 = e^4 and x + y = f^2, with x > 0, y > 0, gcd(x, y) = 1, and e, f > 0, thereby yielding a primitive Pythagorean triangle (x, y, z) where the hypotenuse z = e^2 is a perfect square and the sum of the legs x + y is also a perfect square, distinct from the two previously known primitive solutions.

Background

The paper studies Fermat’s problem of finding a Pythagorean triangle (x, y, z) such that both the hypotenuse z and the sum of the legs x + y are perfect squares. This is encoded by the simultaneous Diophantine equations x^2 + y^2 = e^4 and x + y = f^2 for positive integers x, y, e, f with gcd(x, y) = 1.

The author reviews classical methods and Pell-type equations, presents constructions that yield infinitely many triangles where either the hypotenuse or the sum of the arms is square, and provides an algorithmic framework intended to generate primitive solutions to the full simultaneous system.

Two positive primitive solutions are documented: the classical one attributed to Fermat and another solution previously found by G. Jacob Martens. The author attempted to compute a further (third) positive primitive solution using the proposed method but reports an inability to complete the necessary calculations, leaving the explicit construction unresolved.