Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation

This paper introduces a novel class of prime-generating quadratic polynomials defined by $f_{Z,k,H}(n)=n^2-(2Zk-1)n+\frac{(2Zk-1{)}^2+H}{4}$, where $Zk\in {\mathbb{Z}}_{\ge 0}$ and $H$ belongs to the set of Heegner numbers. This form is closely related to the Euler-Rabinowitsch polynomials through specific substitutions. The structure enables algebraic tuning for prime-rich outputs and provides deeper insight into the impact of Heegner numbers on prime distribution. Using tools such as the Bateman-Horn conjecture and prime-counting functions, we demonstrate that this family can be optimized to generate a high density of primes. This work offers new directions for research in analytic number theory and potential applications in cryptography and signal processing.