Related papers: Properties of Higher-Order Prime Number Sequences
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…
The appearance of primes in a family of linear recurrence sequences labelled by a positive integer $n$ is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in $n$)…
We present a new, elementary, dynamical proof of the prime number theorem.
A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.
In their recent paper, Rosen, Takeyama, Tasaka, and Yamamoto constructed recurrent sequences providing a decomposition law of primes in a Galois extension. In this paper, we reconstruct their sequences via representation theory of finite…
V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
This paper investigates randomness properties of sequences derived from Fibonacci and Gopala-Hemachandra sequences modulo m for use in key distribution applications. We show that for sequences modulo a prime a binary random sequence B(n) is…
This paper presents algorithms for calculating the quadratic character and the norms of prime ideals in the ring of integers of any quadratic field. The norms of prime ideals are obtained by means of a sieve algorithm using the quadratic…
We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the…
We survey the classical results on the prime number theorem
74 new integer sequences are introduced in number theory, and for each of them is given a characterization, followed by open problems. each one a general question: how many primes each sequence has.
Bertrand's Postulate ensures existence of prime $p$ between $n$ and $2n$, $n$ an integer $\geq 2$ and the sieve of Eratosthenes, a very simple ancient algorithm, generates all prime numbers up to any given limit. Combining the above two, in…
We improve lower bounds on the $k$th-order nonlinear complexity of pseudorandom sequences over finite fields and we establish a probabilistic result on the behavior of the $k$th-order nonlinear complexity of random sequences over finite…
We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…
Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…
We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the…
We found a regularity of the behavior of primes that allows to represent both prime and natural numbers as infinite matrices with a common formation rule of their rows. This regularity determines a new class of infinite cyclic groups that…
The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer $k$ there exists a prime number that can be represented as…