Related papers: The Set of Prime Numbers
We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability q) or join an already existing cluster with a…
By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).
Suppose that P is an infinite set of primes such that P = A + B + C, where A,B,C are sets with at least two elements. We show that if P(x) > c x/log^d x (where P(x) = the number of elements of P that are <= x), and if A,B,C is a "regular"…
The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. In this paper, first we state and prove a…
Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci-…
We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions…
We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We…
The sequence of the primes $p$ for which a variety over $\mathbb{Q}$ has no $p$-adic point plays a fundamental role in arithmetic geometry. This sequence is deterministic, however, we prove that if we choose a typical variety from a family…
In two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry {\it win-loss}. We present in…
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 prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of…
In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic…
The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…
We prove that it is always possible to find a permutation $p$ on the set $\{1,...,n\}$ such that $c+p(c)$ is prime for all $c \in \{1,...,n\}.$
It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…
For $n=1,2,3,\ldots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\root n\of{S_n/n}\ (n=1,2,3,\ldots)$ is strictly decreasing, and moreover the sequence $a_{n+1}/a_n\ (n=10,11,\ldots)$ is strictly…
We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.
Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…
Motivated by real-world machine learning applications, we analyze approximations to the non-asymptotic fundamental limits of statistical classification. In the binary version of this problem, given two training sequences generated according…
In recent years, computer simulations are playing a fundamental role in unveiling some of the most intriguing features of prime numbers. In this work, we define an algorithm for a deterministic walk through a two-dimensional grid that we…