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In this set of three companion manuscripts/articles, we unveil our newresults on primality testing and reveal new primality testing algorithmsenabled by those results. The results have been classified (and referred to) aslemmas/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((logN)^2) ; when an explicit value of a Quadratic Non Residue (QNR) modulo-N isavailable (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, whichreveals close connections between the state-of-the-art Miller-Rabin method andthe renowned Euler-Criterion. This Lemma, together with the Baseline PrimalityConjecture enables a synergistic fusion of Miller-Rabin iterations and ourmethod(s), resulting in hybrid algorithms that are substantially better thantheir components. Next, we illustrate how the requirement of an explicit valueof a QNR can be circumvented by using relations of the form: Polynomial(x) modN = 0 ; whose solutions implicitly specify Non Residues modulo-N. We thendevelop a method to derive low-degree canonical polynomials that togetherguarantee implicit Non Residues modulo-N ; which along with the GeneralizedPrimality Conjectures enable algorithms that achieve a worst case deterministicpolynomial complexity = O( (log N)^3 polylog(log N)) ; unconditionally ; forany/all values of N.  In Part/Article 2 , we present substantial experimental data that corroborateall the conjectures. No counter example has been found.  Finally in Part/Article 3, we present analytic proof(s) of the BaselinePrimality Conjecture that we have been able to complete for some special cases.

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