A Dynamical Proof of the Prime Number Theorem

素数无限性的另一种拓扑学证明

Another Topological Proof of the Infinitude of Prime Numbers

本文通过引入一种新拓扑，为素数无限性提供了新的拓扑学证明，并刻画了素数任意非空子集的无限性。

New proof of Tauberian theorem of Laplace Transform and Prime Number Theorem

We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.

An elementary proof of uniqueness of Markoff numbers which are prime powers

We present a very elementary proof of the uniqueness of Markoff numbers which are prime powers or twice prime powers, in the sense that it uses neither algebraic number theory nor hyperbolic geometry.

Reduction of polynomial dynamical systems modulo primes

We study the algebraic dynamical systems generated by triangular systems of rational functions and estimate the height growth of iterations generated by such systems. Further, using a result on the reduction modulo primes of systems of multivariate polynomials over the integers, we study the periodic points and the intersection of orbits of such dynamical systems over finite fields. 1

An alternative proof of the Dirichlet prime number theorem

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative proof of it based on a previous result of us. Also we get an estimation of the prime counting function in the special cases.

Representing Primes as x^2 + 5y^2: An Inductive Proof that Euler Missed

We present an elementary inductive proof which Euler could have obtained, for the corresponding result as the title indicates, had he refined a bit his proof for Fermat's assertion on representing primes as two squares.

On Littlewood's proof of the prime number theorem

In this note we examine Littlewood's proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the non-vanishing of Riemann's zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

A Quixotic Proof of Fermat's Two Squares Theorem for Prime Numbers

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two squares.

On the Dynamical System Generated by the Möbius transformation at Prime Times

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map $x \mapsto (ax + b)/(cx+d)$ over a finite field of $p$ elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences.

A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin's form: $$ π(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev functions as in conventional analytic methods, this new approach uses contour-integration method to analyze Riemann's prime counting function $J(x)$, which only differs from $π(x)$ by $\mathcal O(\sqrt x/\log x)$.

An alternative proof of Sylvester's theorem and variations for more primes

This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger versions of Sylvester's theorem and Bertrand's postulate with more primes, as well as an approach to get more results in this direction.

Harbingers of Artin's Reciprocity Law. I. The Continuing Story of Auxiliary Primes

In this article we present the history of auxiliary primes used in proofs of reciprocity laws from the quadratic to Artin's reciprocity law. We also show that the gap in Legendre's proof can be closed with a simple application of Gauss's genus theory.

A New Proof of Vinogradov's Three Primes Theorem

We give a new proof of Vinogradov's three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of L-functions, either explicitly or implicitly. Our proof uses instead a transference principle, the idea of which was first developed by Green. To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.

素数定理的一个初等 Tauber 证明

An elementary Tauberian proof of the Prime Number Theorem

本文用初等实分析给出素数定理的 Tauber 证明，避免 zeta 函数解析延拓，通过推广 Wiener-Ikehara 定理至 Laplace 变换实现。

On the properties of generalized harmonic and oscillatory numbers. Simple proof of the Prime Number Theorem

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 asymptotic behavior. On the basis of these identities a simple proof of the Prime Number Theorem is represented.

A short proof of the multidimensional Szemerédi theorem in the primes

Tao conjectured that every dense subset of $\mathcal{P}^d$, the $d$-tuples of primes, contains constellations of any given shape. This was very recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler. Here we give a simple proof using the Green-Tao theorem on linear equations in primes and the Furstenberg-Katznelson multidimensional Szemerédi theorem.

The Prime Number Theorem via Karamata's proof of the Landau-Ingham Tauberian theorem

This is an exposition, in 12 pages including all prerequisites and a generalization, of Karamata's little known elementary proof of the Landau-Ingham Tauberian theorem, a result in real analysis from which the Prime Number Theorem follows in a few lines. We believe this to be the most transparent of the elementary proofs of the PNT.

关于利用切比雪夫 Theta 函数的阶估计证明素数定理

On the proof of the Prime Number Theorem using Order Estimates for the Chebyshev Theta Function

本文研究 Selberg 和 Erdős 对素数定理的初等证明，重点推导切比雪夫 Theta 函数的阶估计，并分析辅助函数 ρ(x) 的渐近性质，以建立主要结果的等价表述。

An elemetary proof of an estimate for a number of primes less than the product of the first $n$ primes

Let $α$ be a real number such that $1< α<2$ and let $x_0=x_0(α)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{α-1} -\fracπ{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer $n>x_0$ there exist at least $[n^α]$ primes between the $(n+1)$th prime and the product of the first $n+1$ primes. In particular, we establish a recent Cooke's result which asserts that for each positive integer $n$ there are at least $n$ primes between the $(n+1)$th prime and the product of the first $n+1$ primes. Our proof is based on an elementary counting method (enumerative argument…

New Results on Primes from an Old Proof of Euler's

In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers. Our short paper uses a simple modification of Euler's argument to obtain new results about the distribution of prime factors of sets of integers, including a weak one-sided Tschebyshev inequality. An example shows that there cannot be a prime number theorem in this situation, or even a pair of Tschebyshev inequalities, but it would be very interesting to know if a one-sided Tschebyshev inequality holds.

Yet another Proof of an old Hat

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab + cd with min(a, b) > max(c, d) of two ordered products. This gives a new proof Fermat's Theorem expressing primes of the form 1 + 4N as sums of two squares 1 .

An L-function free proof of Hua's Theorem on sums of five prime squares

We provide a new proof of Hua's result that every sufficiently large integer N\equiv 5(mod\,24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of $L$-functions. Here, we present a proof based on the transference principle first introduced by Green. Using a sieve theoretic approach similar to the work by Shao, we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudorandomness of the majorant of the characteri…

A simple proof of the Markoff conjecture for prime powers

We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a,b,c), where a, b, and c are in increasing order, holds whenever $c$ is a prime power.

Prime Factors of Dynamical Sequences

Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference x_{n+1}-x_n has a primitive prime factor for all sufficiently large n. This result provides a new proof of the infinitude of primes for each rational function f of degree at least 2.

Orbits of Polynomial Dynamical Systems Modulo Primes

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly …

A new elementary proof of the Prime Number Theorem

Let $Ω(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(Ω(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(Ω(n))+\mathrm{o}_{N\to\infty}(1). \] This yields a new elementary proof of the Prime Number Theorem.

关于素数次数不可解方程

On unsolvable equations of prime degree

本文基于Kronecker定理，通过修正Weber证明中的错误，给出一个简洁的证明，用于构造整数系数奇素数次数不可解方程的例子。

Valuations, arithmetic progressions, and prime numbers

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

An induction property for prime counting functions

We provide an elementary proof of an asymptotic formula for prime counting functions. As a minor application we give a new reduction of the proof of Chebotarëv's density theorem to the cyclic case.

动力序列稳定素数的稀疏性

Sparsity of stable primes for dynamical sequences

本文研究多项式动力序列的稳定素数密度，证明正密度稳定素数会迫使其动力 Galois 群近可解，并推断此类多项式只能由线性函数、单项式和 Dickson 多项式复合而成。

Reductions Modulo Primes of Systems of Polynomial Equations and Algebraic Dynamical Systems

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the algebraic closure of the field with $p$ elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture.

Towards a bifurcation theory for perturbed monomial dynamical systems modulo a prime

We investigate perturbed monomial dynamical system over $\mathbb{F}_p$ given by iterations of $x\mapsto x^n+c\bmod{p}$, where $c\in \mathbb{F}_p$. Instead of study the systems one at a time we study all of them at the same time. The complex distibution of periodic points is visualized in the so called Periodic Point Diagram, which can be seen as a discrete version of the classical Bifurcation Diagram. We also prove some general results about the distribution of periodic points. We end the article with a conjecture about the total number of periodic points.

Prime divisors of sequences of integers

In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical result which says that a non-degenerate linear recurrence sequence of integers of order k>1 has infinitely many prime divisors.

Proof of Legendre's Conjecture

Legendre's conjecture states that there exists a prime between $n^2$ and $(n+1)^2$, for every positive integer $n$. Here I prove that for sufficiently large $n$, there is a prime number between $n^2$ and $(n+1)^2$. The proof relies on the idea of counting the maximum power, $o_p(n)$ of a prime $p\leq n$ such that $p^{o_p(n)}||n$.

素数定理的动力学推广以及加法与乘法半群作用的不交性

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions

本研究建立两个蕴含多项乘性数论经典结果的遍历定理，将Sarnak的Möbius不交性猜想重构于新动力学框架，提出聚焦加乘半群作用不交性的猜想扩展并证明其若干特例。

仅使用素数同源性的Stéphanois定理

The Stéphanois theorem with only prime isogenies

本文改进了Stéphanois定理的证明，仅使用素数度的模多项式，简化了证明工具并限制了素数的算术序列。

A new proof of Dirichlet's theorem concerning prime numbers in arithmetic progressions

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime number counting function to provide a new proof of Dirichlet's theorem. We show that if $a\in \mathbb{N},k\in \mathbb{N},a_{k}=(a,a+1,...,a+k-1) $ and $A=\left\{ p_{1},p_{2},...,p_{n}\right\} $, a finite set of primes. Then the number of components of $a_{k}$ that are divisible by some prime in $A$ is less than or equal to $$ \sum\limits_{\substack{ d|P(A)\\ d>1}…

Using Dynamical Systems to Construct Infinitely Many Primes

Euclid's proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics. After acceptance Soundararajan noted the beautiful and fast converging formula: $$ τ= a^{1/(d-1)} x_0 \cdot \lim_{n\to \infty} \prod_{m=1}^n \left(\frac{x_m}{ax_{m-1}^d} \right)^{1/d^m} $$

Primes, Pi, and Irrationality Measure

A folklore proof of Euclid's theorem on the infinitude of primes uses the Euler product and the irrationality of $ζ(2) = π^2/6$. A quantified form of Euclid's Theorem is Bertrand's postulate $p_{n+1} < 2p_n$. By quantifying the folklore proof using an irrationality measure for $6/π^2$, we give a proof (communicated to Paulo Ribenboim in 2005) of a much weaker upper bound on $p_{n+1}$.

Fermat Theorems -- Simple Proofs

The Modular Group provides simple proofs of Fermat's representations: X^2+Y^2 for primes congruent to 1 (mod 4) and by X^2+3Y^2 for primes congruent to 1 (mod 3)

A Theorem of Congruent Primes

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter than existing proofs, but the method may be useful in other contexts. The proof of Theorem 1 tracks the set of solutions and this set branches as a binary tree. Conditions set to the theorem restricts the branches so that only one branch is left. Following this branch gives either a solution or a contradiction. In Theorem 1 it leads to a contradiction. The inte…

Development of sieve of Eratosthenes and sieve of Sundaram's proof

We make two algorithms that generate all prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we use two formulas to achieve this development, where all the multiples of prime number 2 are eliminated in the first formula, and all the multiples of prime numbers 2 and 3 are eliminated in the second formula. Using the first algorithm we proof sieve of Sundaram's algorithm, then we improve it to be more efficient prime generating algorithm. We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm…

A Corrected Simplified Proof of Chen's Theorem

In 1973, J.-R. Chen showed that every large even integer is a sum of a prime and a product of at most two primes. In this paper, the author indicates and fixes the issues in a simplified proof of this result given by Pan et al.

Linear equations in primes and dynamics of nilmanifolds

We survey some of the ideas behind the recent developments in additive number theory, combinatorics and ergodic theory leading to the proof of Hardy- Littlewood type estimates for the number of prime solutions to systems of linear equations of finite complexity.

Primes, knots and periodic orbits

This is a text written for the Ennio De Giorgi Colloquio volume. It covers analogies between algebraic number theory and knot theory, analogies between analytic number theory and certain dynamical systems, and a report on our construction of dynamical systems for arithmetic schemes which realize some of these analogies.

关于保持多项式动态不可约性的素数频率

On the frequency of primes preserving dynamical irreducibility of polynomials

本文针对算术动力学中多项式迭代模素数不可约性的素数密度问题，通过使用 Selberg 筛法改进了以往基于 Brun 筛法的结果，在部分情况下实现了显著的定量提升。

Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis

For n>1, let G(n)=σ(n)/(n log log n), where σ(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erdős's results.

Sifting for small primes from an arithmetic progression

In this work and its sister paper [5] we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dipense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and reelied on in earlier proofs.

The Prime Number Theorem as a Capstone in a Complex Analysis Course

We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for any instructor who seeks to use the prime number theorem for a series of capstone lectures, a scaffold for a series of guided exercises, or as a framework for an inquiry-based course. We require almost no knowledge of number theory, for our aim is to make a complete proof of the prime number theorem widely accessible to complex analysis instructors. In particular, we highlight the potential pitfalls and subtleties t…

Folkman定理与素数

Folkman's theorem and the primes

本文通过Folkman定理（或Hindman定理）给出了素数无限性的两个新证明，扩展了利用Ramsey型定理推导素数无限性的文献。

Almost all primes satisfy the Atkin-Serre conjecture and are not extremal

Let $f(z)=\sum_{n=1}^{\infty} a_f(n)e^{2πi n z}$ be a non-CM holomorphic cupsidal newform of trivial nebentypus and even integral level $k\geq 2$. Deligne's proof of the Weil conjectures shows that $|a_f(p)|\leq 2p^{\frac{k-1}{2}}$ for all primes $p$. We prove for 100% of primes $p$ that $2p^{\frac{k-1}{2}}\frac{\log\log p}{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor$. Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin-Serre conjecture is satisfied for 100% of primes, and the upper bound shows that $|a_f(p)|$ is as l…

区间$ [15x,16x] $内素数的存在性——一个完全初等的证明

Existence of primes in the interval $ [15x,16x] $ -- An entirely elementary proof --

本文基于Erdös等人的素数区间证明思路，给出了k≤15时区间[kx,(k+1)x]内对充分大x至少含N个素数这一命题的完全初等证明，其中k≥5的情形采用了新思路。

Fermat's Last Theorem, Schur's Theorem (in Ramsey Theory), and the Infinitude of the Primes

Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n=3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.

On Proving of Diophantine Inequalities with Prime Numbers by Evaluations of the Difference between Consecutive Primes

Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.