F. J. M. Barning

F. J. M. Barning gave a matrix-based generation process for primitive Pythagorean triples. His 1963 Mathematisch Centrum report verifies the initials and presents three unimodular matrices generating all such triples without repetition, with a fixed leg-order convention.^[ref]

The transformations preserve the equation $x^2+y^2=z^2$. Starting from $(4,3,5)$, with the even leg first, repeated transformations build a branching tree of solutions. The method turns a Diophantine equation into a recursive enumeration algorithm with a uniqueness theorem.^[ref]