On the Existence and Frequency Distribution of the Shell Primes

TL;DR

This paper investigates the existence and distribution of shell primes, defined via a p-dimensional half-shell formula, and explores their properties using the Euler zeta function without sieving, revealing new insights into prime generation.

Contribution

It introduces the concept of shell primes and analyzes their frequency distribution, applying the Euler zeta function in novel non-sieving contexts to understand prime occurrence.

Findings

Shell primes are characterized by the formula n^p - (n-1)^p with prime p.

Application of the Euler zeta function to shell primes reveals patterns in their distribution.

The study suggests potential methods for predicting prime counts from polynomial functions.

Abstract

This research presents the results of a study on the existence and frequency distribution of the shell primes defined herein as prime numbers that result from the calculation of the "half-shell" of an p-dimensional entity of the form $n^p-(n-1)^p$ where power $p$ is prime and base $n$ is the realm of the positive integers. Following the introduction of the shell primes, we will look at the results of a non-sieving application of the Euler zeta function to the prime shell function as well as to any integer-valued polynomial function in general which has the ability to produce prime numbers when power $p$ is prime. One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no known way to sieve composite numbers out of the product term in this famous equation. Such would be case when an infinite series of numbers to be analyzed are calculated by a polynomial expression that yields successively increasing positive integer values and which has as its input domain the positive integers themselves. In such cases there may not be an intuitive way to eliminate the composite terms from the product term in the Euler zeta function equation by either scaling a previous prime number calculation or by employing predictable values of the domain of the function which would render outputs of the polynomial prime. So the best one may be able to hope for in these cases is to calculate some value to be added or subtracted from unity in the numerator above the product term in the Euler Zeta function to make both sides of that equation equal with the expectation that that value could be used to predict the number of prime numbers that exist as outputs of the polynomial function for some limit less than or equal to x of the input domain.