Notes on Prime Numbers, Their Numerical Statistics & Patterns I: Modular Arithmetic and the Eight Fold Period 24 Model

Abstract

We have not yet obtained a simple, complete and accurate method for determining if an integer number specifically an odd number that does not digitally sum to 3,6,9, whether it is prime, and that does not involve trial & error and/or as by a sieving method or factoring. This after thousand(s) of years of searching for such. Alternatively great strides are made towards understanding the behavior of numbers, fields of numbers obtained by some method or another, statistics probabilities & entropies of such numbers, & so forth, & it seems that it is assumed that a better method (but not 'full') for determining primality will appear as an afterthought or byproduct. In this letter we wish to revitalize the interest of the intrepid professional & amateur mathematician & scientist towards seeking the 'full' or 'complete' method of determining primality, or rather towards obtaining the method by which a prime odd integer can be determined completely, simply and efficiently & with applicability to small to very large numbers with a minimum of effort eschewing supercomputing & other recent methods however by utilizing methods resembling pencil & paper approaches.

The caveat that we wish the 'method(s)' to be efficient and simple means specifically that we are referring to the utilization of methods such as digital sums & digital roots, modulus methods, combinatorial & progressions & recursions patterns easily applied, these connected to geometries, polynomial equations and relationships, discernible patterns in sieves and doodles and spirals & other topologies investigated in one context or another yet often deemed curiosities when viewed from the focused area of number theory as pertaining to identifying primes.

Keywords: prime numbers, twin primes, cousin primes, nearest neighbor, integer numbers, number theory, numerical, statistical distributions, random matrix theory, nonextensive, nonextensive random matrix theory