Conjecture B: Parametric representation of Pythagorean primes via (4ab−1)(4c−1) − 4ac

Establish that every Pythagorean prime p (i.e., p ≡ 1 mod 4) can be expressed as p = (4ab − 1)(4c − 1) − 4ac, where a, b, and c are natural numbers.

Background

Complementing Type (A), the paper’s Type (B) reduction gives a necessary and sufficient parametrization of primes p ≡ 1 (mod 4) based on variables a, u, and a divisor d of a^2, motivating the structural form stated in Conjecture B.

Conjecture B proposes that all Pythagorean primes can be written as (4ab−1)(4c−1) − 4ac with a,b,c ∈ N, a form supported by extensive empirical checks. If Conjecture B holds, it would ensure solvability via the Type (B) pathway and, together with Conjecture A, would imply the Erdős–Straus conjecture.