Analytic Proof of the recent Baseline Primality Conjecture
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2021-12-12
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Attribution 4.0 International (CC BY 4.0)
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Abstract
This document presents an analytic proof of the Baseline Primality Conjecture (BPC) that was recently
unveiled in [1, Part I]. The BPC identifies a new small set of conditions that are sufficient to decide
the primality of any input integer N under test (see Section 2 for the exact statement of the BPC
in the original form using algebraic integers; and Section 3 for an equivalent polynomial domain
reformulation).
The practical significance of the BPC is that it directly leads to ultra low complexity primality testing
algorithms, wherein the number of bit-operations required is essentially a quadratic function of the bit length of the input N [1]. More specifically, the Baseline Primality Result (BPR) demonstrates that after
an/any integer in the closed interval [2,N −2] which is a Quadratic Non Residue (QNR) modulo-N
is found; exactly 2 (Two, which is a small O(1) constant, independent of the bit-length of the input N)
specific modular exponentiations are sufficient to determine whether N is a composite or a prime.
The BPC was (and to this day continues to be) extensively tested numerically.1
Additionally, analytic proofs of the BPC for several specific forms of the input N were also presented
in [1], wherein the BPR was first unveiled. However, at the time of the original publication [1], we were
not able to complete a general analytic proof of the BPC that covered all possible cases (i.e., forms) of the
input N.
We have now completed that vital task by developing a general analytic proof of the BPC using its
polynomial domain reformulation. A concise presentation of that analytic proof is the main and
narrow focus as well as the main new contribution of this article. An auxiliary contribution is a clear
and precise explanation of the intuition behind our approach and the illustration of how it leads to the
new theoretical results developed in [1].