Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation

TL;DR

This paper introduces a new class of prime-generating quadratic polynomials linked to Heegner numbers, offering insights into prime distribution and potential applications in cryptography.

Contribution

It defines a novel prime-generating polynomial family based on Heegner numbers, expanding the understanding of algebraic structures influencing prime distribution.

Findings

Polynomial family can be optimized for high prime density

Connection established between Heegner numbers and prime-rich polynomials

Potential applications in cryptography and signal processing

Abstract

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 $Zk \in \mathbb{Z}_{\geq 0}$ 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.