Elaborating the Condition in Fermat’s Two Square Theorem

It is a well-known result that prime numbers congruent to 1 modulo 4 are representable as a sum of two squares. A recent article shows that the number of representations can imply the primality of square-free integers congruent to 1 modulo 4. However, the authors did not clarify the distinctness of the construction of representations in the proof of the main theorem. We provide a complementary proof to clarify this construction and also present an alternative proof based on Jacobi’s two square theorem.