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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.02821 |
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| _version_ | 1866915426710061056 |
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| author | Kumaresan, Sudarshan Kumari, Shipra Mishra, Neha |
| author_facet | Kumaresan, Sudarshan Kumari, Shipra Mishra, Neha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation Kumaresan, Sudarshan Kumari, Shipra Mishra, Neha Number Theory 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. |
| title | Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.02821 |