Related papers: Describe Prime number gaps pattern by Logistic map…
We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness…
A continuously measured quantum system with multiple jump channels gives rise to a stochastic process described by random jump times and random emitted symbols, representing each jump channel. While much is known about the waiting time…
For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\mathbb{Z}/P\mathbb{Z}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a…
The prime numbers look like a randomly chosen sequence of natural numbers, but there is still no strict theory to determine 'Randomness'. In these years, cryptography has developed a battery of statistical tests for randomness. In this…
Research on the distribution of prime numbers has revealed a dual character: deterministic in definition yet exhibiting statistical behavior reminiscent of random processes. In this paper we show that it is possible to use an image-focused…
In this article, a relation between a gap $d_{k}$ and divisors of composite numbers between $p_{k}$ and $p_{k+1}$ is established.
We treat three cubic recurrences, two of which generalize the famous iterated map $x \mapsto x (1-x)$ from discrete chaos theory. A feature of each asymptotic series developed here is a constant, dependent on the initial condition but…
We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $\psi(x;q,a)$ and $\psi(x;q,b)$ are simultaneously larger than their asymptotic main terms, as well as a method for…
in the last decade, studies of chaotic system are more often used for classical choatic system than for quantum chaotic system, there are many ways of observing the chaotic system such us analyzing the frequency with Fourier transform or…
The pattern dynamics of the one-way coupled logistic lattice which can serve as a phenomenological model for open flow is investigated and shown to be extremely rich. For medium and large coupling strengths, we find spatially periodic,…
We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$…
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…
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…
We prove that the set of normalized differences between primes, defined as $S = \{(p-q)/(p+q) : p > q \text{ are primes}\}$, is dense in the open unit interval $(0,1)$. Our proof provides an explicit construction algorithm with quantitative…
In this paper we show how the cross-disciplinary transfer of techniques from Dynamical Systems Theory to Number Theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics…
Pairs of numerically computed trajectories of a chaotic system may coalesce because of finite arithmetic precision. We analyse an example of this phenomenon, showing that it occurs surprisingly frequently. We argue that our model belongs to…
To a unitary matrix U we associate a doubly stochastic matrix M by taking the modulus squared of each element of U. To study the connection between onset of quantum chaos on graphs and ergodicity of the underlying Markov chain, specified by…
We study the logistic map $f(x)=\lambda x(1-x)$ on the unit square at the chaos threshold. By using the methods of symbolic dynamics, the information content of an orbit of a dynamical system is defined as the Algorithmic Information…
In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…
We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness…