Near NP-Completeness for Detecting p-adic Rational Roots in One Variable

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus improve the best previous complexity upper bound of EXPTIME. We also prove an unconditional complexity lower bound of NP-hardness with respect to randomized reductions for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss how our results complement analogous results over the real numbers.Comment: 8 pages in 2 column format, 1 illustration. Submitted to a conferenc

BiEntropy, TriEntropy and Primality

The order and disorder of binary representations of the natural numbers < 2^8 is measured using the BiEntropy function. Significant differences are detected between the primes and the non primes. The BiEntropic prime density is shown to be quadratic with a very small Gaussian distributed error. The work is repeated in binary using a monte carlo simulation for a sample of the natural numbers < 2^32 and in trinary for all natural numbers < 3^9 with similar but cubic results. We find a significant relationship between BiEntropy and TriEntropy such that we can discriminate between the primes and numbers divisible by six. We discuss the theoretical underpinnings of these results and show how they generalise to give a tight bound on the variance of Pi(x) - Li(x) for all x. This bound is much tighter than the bound given by Von Koch in 1901 as an equivalence for proof of the Riemann Hypothesis. Since the primes are Gaussian due to a simple induction on the binary derivative, this implies that the twin primes conjecture is true. We also provide absolutely convergent asymptotes for the numbers of Fermat and Mersenne primes in the appendices.Comment: 18 Pages, 12 Colour Figures, 10 Tables. Minor updates & typos. Supplementary materials now available at https://doi.org/10.6084/m9.figshare.11743749. Empirical results first presented at ANPA 40, University of Liverpool, 11th August 2019. Theoretical Basis first presented at PANPA 2019 meeting, Anstruther, Fife, Scotland, 21st August 201