Related papers: Properties of Higher-Order Prime Number Sequences
We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.
We suggest other models of sieve generated sequences like the Sieve of Eratosthenes to explain randomness properties of the prime numbers, like the twin prime conjecture, the lim sup conjecture, the Riemann conjecture, and the prime number…
In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
We propose a multi-scale analysis method for studying arithmetic properties of integer sets, such as primality. Our approach organizes information through a hierarchy of nested sequences, where each level enables a hierarchical expression…
We present a variety of prime-generating constructions that are based on sums of primes. The constructions come in all shapes and sizes, varying in the number of dimensions and number of generated primes. Our best result is a construction…
For every integer $n\ge 1$ let $a_n$ be the smallest positive integer such that $n+a_n$ is prime. We investigate the behavior of the sequence $(a_n)_{n\ge 1}$, and prove asymptotic results for the sums $\sum_{n\le x} a_n$, $\sum_{n\le x}…
We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their…
An elementary method for computing various prime sequences using the sequence of Farey sequences is described.
This work is meant to demonstrate new class of prime numbers -- cyclic prime numbers, that can be derived from any prime number at certain numeric systems. Cyclic prime numbers are also related to the cyclic numbers and full reptend prime…
We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we…
This paper presents a distinctive prime detection approach. This method use GM-(n+1) sequences to effectively eliminate complex numbers. The sequences, which consist of odd a number of (n+1), exclude all components except for the initial…
We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from…
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.
A modified Lagrange Polynomial is introduced for polynomial extrapolation, which can be used to estimate the equally spaced values of a polynomial function. As an example of its application, this article presents a prime-generating…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence $C_n = np_n - \sum_{k \leq n}p_k$, $n \geq 1$, involving the prime numbers.
Based on Euclid's algorithm, we find a kind of special sequences which play an interesting role in the study of primes. We call them W Sequences. They not only ties up the distribution of primes in short interval but also enables us to give…
A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the…