  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.
