PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion Manuscripts

TL;DR

This paper introduces new deterministic primality testing algorithms with low complexity, leveraging explicit and implicit non-residues, and provides experimental and partial analytic validation of the conjectures involved.

Contribution

It proposes novel primality tests based on non-residues, combining existing methods with new algorithms, achieving polynomial complexity and supporting conjectures with experimental data.

Findings

Deterministic primality test with complexity O((log N)^2) for most odd integers.

Hybrid algorithms outperform individual methods by integrating Miller-Rabin and new approaches.

Experimental data supports conjectures, with partial proofs for special cases.

Abstract

In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as lemmas/corollaries/claims whenever we have complete analytic proof(s); otherwise the results are introduced as conjectures. In Part/Article 1, we start with the Baseline Primality Conjecture~(PBPC) which enables deterministic primality detection with a low complexity = O((log N)^2) ; when an explicit value of a Quadratic Non Residue (QNR) modulo-N is available (which happens to be the case for an overwhelming majority = 11/12 = 91.67% of all odd integers). We then demonstrate Primality Lemma PL-1, which reveals close connections between the state-of-the-art Miller-Rabin method and the renowned Euler-Criterion. This Lemma, together with the Baseline Primality Conjecture enables a synergistic fusion of Miller-Rabin iterations and our method(s), resulting in hybrid algorithms that are substantially better than their components. Next, we illustrate how the requirement of an explicit value of a QNR can be circumvented by using relations of the form: Polynomial(x) mod N = 0 ; whose solutions implicitly specify Non Residues modulo-N. We then develop a method to derive low-degree canonical polynomials that together guarantee implicit Non Residues modulo-N ; which along with the Generalized Primality Conjectures enable algorithms that achieve a worst case deterministic polynomial complexity = O( (log N)^3 polylog(log N)) ; unconditionally ; for any/all values of N. In Part/Article 2 , we present substantial experimental data that corroborate all the conjectures. No counter example has been found. Finally in Part/Article 3, we present analytic proof(s) of the Baseline Primality Conjecture that we have been able to complete for some special cases.