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June 6, 2026Open Access

A Parametric Family of Primes p = km(m+1) + ε + 2kq: Heuristic Laws, Conditional Theorems, and Unconditional Primality Certificates

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Authors

HBHASSANE BAKKAOUI

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Overview

This preprint uncovers parametric integer families for primes and offers implications for primality testing.

Key Points

  • The aim is to study a specific family of primes defined by a parametric formula and derive results regarding their properties.
  • Systematic exploration of primes expressed as p = km(m+1) + ε + 2kq.
  • Derivation of results leveraging classical and conditional number theory hypotheses.
  • Numerical validation of heuristic laws for large values.
  • Proved that for any prime ℓ dividing 2k, q_min is uniformly distributed modulo ℓ with an error of O(N^{-1/2}).
  • An unconditionally certified prime of 29,998 decimal digits was achieved using Pocklington–Lehmer certificates.
  • Established absence of spectral correlations between functions Q(r) and the zeros of ζ(s), supported by multiple statistical tests.

Cite This Study

HASSANE BAKKAOUI (2026) studied this question.

synapsesocial.com/papers/6a23bbeb71a5da9775e774b0https://doi.org/10.5281/zenodo.20544946
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