Related papers: Dynamical Sieve of Eratosthenes
For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x,…
Factorization is the most fundamental way to determine if a number $n$ is prime or composite. Yet, this approach becomes impracticable when considering large values of $n$, a difficulty that is exploited by cryptographic protocols. We…
Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for…
In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some…
The original continuous-time "goldfish" dynamical system is characterized by two neat formulas, the first of which provides the $N$ Newtonian equations of motion of this dynamical system, while the second provides the solution of the…
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…
We call a set of positive integers closed under taking unitary divisors a unitary ideal. It can be regarded as a simplicial complex. Moreover, a multiplicative arithmetical function on such a set corresponds to a function on the simplicial…
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation…
A discrete map based on the sum of an integer's distinct primes factors and the sum of its other factors is defined and its iteration is studied.
A representation number is a function which expresses the number of ways an integer can be written as a sum of elements of chosen sets. One of the oldest number-theoretic results on representation numbers is Fermat's theorem which says that…
We discuss analogies between the etale site of arithmetic schemes and the algebraic topology of dynamical systems. The emphasis is on Lefschetz numbers. We also discuss similarities between infinite primes in arithmetic and fixed points of…
Can we do arithmetic in a completely different way, with a radically different data structure? Could this approach provide practical benefits, like operations on giant numbers while having an average performance similar to traditional…
We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least.…
In this paper, we introduce a family of discrete rectangular uniform distributions on the natural numbers-referred to as orthogonal dice-characterized by the property that their means equal their variances. These distributions arise…
Let $x$ be a positive integer. We give an asymptotic formula for the number of primes in the set $\{\fl{x/n}, 1 \le n \le x\}$ and give some related results.
We investigate by means of a simple theoretical model the emergence of prime numbers as life cycles, as those seen for some species of cicadas. The cicadas, more precisely, the Magicicadas spend most of their lives below the ground and then…
We investigate four-fermion interactions with $N$-component fermion in Einstein universe for arbitrary space-time dimensions ($2 \leq D<4$). It is found that the effective potential for composite operator $\overline{\psi}\psi$ is calculable…
We investigate the dynamics of kinetically constrained models of glass formers by analysing the statistics of trajectories of the dynamics, or histories, using large deviation function methods. We show that, in general, these models exhibit…
A finite collection $P$ of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members of $P$. We associate with such a tiling a doubly infinite sequence with entries from $P$. The set of…
Making use of covering systems and a theorem of D. Shiu, the first and second authors showed that for every positive integer $k$, there exist $k$ consecutive widely digitally delicate primes. They also noted that for every positive integer…