Related papers: Properties of Higher-Order Prime Number Sequences
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…
Let $\underline{a}$ and $\underline{b}$ be primitive sequences over $\mathbb{Z}/(p^e)$ with odd prime $p$ and $e\ge 2$. For certain compressing maps, we consider the distribution properties of compressing sequences of $\underline{a}$ and…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set…
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of…
The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.
In the course of studies of the measure of chaos for the distribution of the prime numbers among the positive integers N arched structures have been found. It is given a brief description of the fine structure of the positive integers…
In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…
In this paper, we consider properties of coefficients of a generating functions composition, where the outer function is a logarithmic generating function and the inner function is an ordinary generating function with integer coefficients.…
This paper investigates prime and co-prime integer matrices and their properties. It characterizes all pairwise co-prime integer matrices that are also prime integer matrices. This provides a simple way to construct families of pairwise…
We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new…
We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…
In this document, prime numbers are related as functions over time, mimicking the Sieve of Eratosthenes. For this purpose, the mathematical representation is a uni-dimentional time line depicting the number line for positive natural numbers…
We report the first experimental demonstration of prime number sieve via linear optics. The prime numbers distribution is encoded in the intensity zeros of the far field produced by a spatial light modulator hologram, which comprises a set…
There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide…
In this paper we introduce the notion of the $P$-sequences and apply their properties in studying representability of real numbers. Another application of $P$-sequences we find in generating the Prouhet-Tarry-Escott pairs.
In this paper we examine the percolation properties of higher-order networks that have non-trivial clustering and subgraph-based assortative mixing (the tendency of vertices to connect to other vertices based on subgraph joint degree). Our…
Odd numbers can be indexed by the map k(n)=(n-3)/2, n belonging to 2N+3. We first propose a basic primality test using this index function that was first introduced in article (8). Input size of operations is reduced which improves…
The dominant approach to sequence generation is to produce a sequence in some predefined order, e.g. left to right. In contrast, we propose a more general model that can generate the output sequence by inserting tokens in any arbitrary…
In this note we prove an inequality involving primes and the product of consecutive primes.