Pythagorean Divisor Families and Explicit Rational Points on Congruent-Number Elliptic Curves

This manuscript develops a number-theoretic construction connecting divisor-based families of Pythagorean triangles with distinguished rational points on congruent-number elliptic curves. It establishes the underlying correspondence, studies its structural and uniqueness properties, and relates the resulting elliptic-curve families to an elementary characterization of odd primes. The mathematical results are accompanied by a limited discussion of computational applications over finite fields. The implementation-related study concerns two fixed research domains and includes domain qualification and verification of selected elliptic-curve operations.