Conjecture A: Parametric representation of Pythagorean primes via (4ab−1)(4c−1) − 4ab^2/d

Establish that every Pythagorean prime p (i.e., p ≡ 1 mod 4) can be expressed as p = (4ab − 1)(4c − 1) − 4ab^2/d, where a, b, and c are natural numbers and d is a divisor of ab.

Background

The paper derives a reduction of the Erdős–Straus equation for primes into two simpler Diophantine equations corresponding to solution types (A) and (B). Type (A) solutions lead to a parametrization of primes p ≡ 1 (mod 4) using numbers of the form 4w − 1 (denoted Bw), culminating in a family that appears empirically to capture all Pythagorean primes.

Conjecture A encodes this observation as a structural representation: every Pythagorean prime admits the explicit form (4ab−1)(4c−1) − 4ab^2/d with a,b,c ∈ N and d | ab. Validity of Conjecture A would imply solvability of the Type (A) equation and contribute toward proving the ESC.