arXiv subjects
Publications and source records attributed to Michael P. May.
We apply the Inclusion-Exclusion Principle to a unique pair of prime number subsequences to determine whether these subsequences form a small set or a large set and thus whether the infinite sum of the inverse of their terms converges or diverges. In this paper, we analyze the complementary prime number subsequences P(prime) and P(double-prime) as well as revisit the twin prime subsequence P2.
A closed-form solution for the boundary of the flat state of an orthogonal cross section of contiguous surface geometry formed by the intersection of two cylinders of equal radii oriented in dual directions of rotation about their intersecting axes.
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.
In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic selection and elimination of prime number indexes on the real number line.
We reveal a relationship between the prime counting function and an operation performed on a unique subsequence of the primes.
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 on the right-hand side of 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 make future outputs of the polynomial prime. So the best one may be able to hope for in this case is to determine some value to be added or subtracted from unity in the numerator above the product term to make both sides of the equation equal in the hope that that value can 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.
This research paper presents the results of a study on the application of extended analytic continuation to locate the non-trivial zeros of the Riemann Hypothesis.