An infinitude of proofs of the infinitude of primes

An extension of Furstenberg's theorem of the infinitude of primes

The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms of an arithmetic progression whose first term is $a_{1}=m-n+1$ and whose difference is $d=2$. Generalizing this idea, we define new similar product mappings, and we consider new arithmetics that enable us to extend Furstenberg's theorem of the infinitude of primes. We also review the classic conjectures in the new arithmetics. Finally, we make important extensions of the main idea. We see that given any integer sequence, the approach generates an arithmetic on integers and a similar formula to $\mathbb{Z} \setmi…

A fractal proof of the infinitude of primes

This short paper gives another proof of the infinitude of primes by using upper box dimension, which is one of fractal dimensions.

The infinitude of $\mathbb{Q}(\sqrt{-p})$ with class number divisible by $16$

The density of primes $p$ such that the class number $h$ of $\mathbb{Q}(\sqrt{-p})$ is divisible by $2^k$ is conjectured to be $2^{-k}$ for all positive integers $k$. The conjecture is true for $1\leq k\leq 3$ but still open for $k\geq 4$. For primes $p$ of the form $p = a^2 + c^4$ with $c$ even, we describe the 8-Hilbert class field of $\mathbb{Q}(\sqrt{-p})$ in terms of $a$ and $c$. We then adapt a theorem of Friedlander and Iwaniec to show that there are infinitely many primes $p$ for which $h$ is divisible by $16$, and also infinitely many primes $p$ for which $h$ is divisible by $8$ but n…

Infinitude of Primes Using Formal Language Theory

Formal languages are sets of strings of symbols described by a set of rules specific to them. In this note, we discuss a certain class of formal languages, called regular languages, and put forward some elementary results. The properties of these languages are then employed to prove that there are infinitely many prime numbers.

On the Infinitude of Prime Ideals in Dedekind Domains

Let $R$ be an infinite Dedekind domain with at most finitely many units, and let $K$ denote its field of fractions. We prove the following statement. If $L/K$ is a finite Galois extension of fields and $\mathcal{O}$ is the integral closure of $R$ in $L$, then $\mathcal{O}$ contains infinitely many prime ideals. In particular, if $\mathcal{O}$ is further a unique factorization domain, then $\mathcal{O}$ contains infinitely many non-associate prime elements.

欧几里得素数无穷多定理：其证明的历史综述（公元前300年—2022年）以及另一个新证明

Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2022) and another new proof

本文是欧几里得素数无穷多定理证明的历史综述，梳理了公元前300年至2022年的200种证明，分类呈现不同时期、不同方法的证明，还给出了一个新的简洁欧几里得式证明。

费马大定理蕴含欧几里得素数无穷性定理

Fermat's Last Theorem Implies Euclid's Infinitude of Primes

该研究借助费马大定理与舒尔组合定理给出小指数情形下欧几里得素数无穷性定理的新证明，还探讨了多个组合数论结果对该定理的蕴含，揭示其为诸多数学结果的必要条件。

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.

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。

A variant of the Euclid-Mullin sequence containing every prime

We consider a generalization of Euclid's proof of the infinitude of primes and show that it leads to variants of the Euclid-Mullin sequence that provably contain every prime number.

Infinitely Many Primes Using Generating Functions

In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In addition, our proof highlights an interesting use of generating functions.

Primes in arithmetic progressions to large moduli, and shifted primes without large prime factors

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we estimate primes of size $x$ with quadrilinear forms of moduli up to $x^{17/32}$. This extends moduli beyond $x^{11/21}$, recently obtained by Maynard, improving $x^{29/56}$ from well-known 1986 work of Bombieri, Friedlander, and Iwaniec.

利用加权筛法证明孪生素数乘积对数倒数之和的正下界

A Positive Lower Bound for the Sum of Log-Reciprocal Twin Prime Products via Weighted Sieve

本文通过改进加权筛法，证明了孪生素数乘积对数倒数之和的正下界，从而推导出孪生素数对的无限性。

某些芒福德阿贝尔四维簇的无穷多个超奇异素数

Infinitely many supersingular primes for some Mumford's abelian fourfolds

本文将 Elkies 关于实数域上椭圆曲线超奇异素数无穷性的结果推广到 Mumford 族中的某些阿贝尔四维簇及更一般的 Kuga-Satake 阿贝尔簇族，通过研究 Hodge 型 Shimura 簇的局部形变空间和 Shimura 曲线的实点完成证明。

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 $p_{n+1}$.

素数幂域上椭圆曲线的殆素数阶

Almost Prime Orders of Elliptic Curves Over Prime Power Fields

本文研究素数 p 使得 |E(F_{p^l})| / |E(F_p)| 具有有界素因子的无穷性，对 CM 曲线无条件应用大筛法，对非 CM 曲线在 GRH 下应用 Chebotarev 密度定理，并利用向量筛法给出相关殆素数个数的下界。

There are Infinitely Many Perrin Pseudoprimes

This paper proves the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. The result uses ingredients of the proof of the infinitude of Carmichael numbers, along with zero-density estimates for Hecke L-functions.

On Generalized Carmichael Numbers

Given an integer $k$, define $C_k$ as the set of integers $n > \max(k,0)$ such that $a^{n-k+1} \equiv a \pmod{n}$ holds for all integers $a$. We establish various multiplicative properties of the elements in $C_k$ and give a sufficient condition for the infinitude of $C_k$. Moreover, we prove that there are finitely many elements in $C_k$ with one and two prime factors if and only if $k>0$ and $k$ is prime. In addition, if all but two prime factors of $n \in C_k$ are fixed, then there are finitely many elements in $C_k$, excluding certain infinite families of $n$. We also give conjectures abou…

Generalizing Ruth-Aaron Numbers

Let $f(n)$ be the sum of the prime divisors of $n$, counted with multiplicity; thus $f(2020)$ $= f(2^2 \cdot 5 \cdot 101) = 110$. Ruth-Aaron numbers, or integers $n$ with $f(n)=f(n+1)$, have been an interest of many number theorists since the famous 1974 baseball game gave them the elegant name after two baseball stars. Many of their properties were first discussed by Erdös and Pomerance in 1978. In this paper, we generalize their results in two directions: by raising prime factors to a power and allowing a small difference between $f(n)$ and $f(n+1)$. We prove that the number of integers up t…

Landau素数理想定理的一个新的初等证明及相关结果

A New Elementary Proof of Landau's Prime Ideal Theorem, and Associated Results

本文给出Landau素数理想定理的新初等证明，推广了Richter的素数定理证明，并包含素数理想计数函数等分布的相关结果。

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.

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.

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.

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.

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.

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.

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 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.

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.

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 $p^{o_p(n)}||n$.

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}…

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.

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.

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。

Mertens's theorem for splitting primes and more

Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the conclusion in a direct and elegant manner". Hardy's proof is also quite adaptable, and it is readily combined with well-known results from prime number theory. We demonstrate this by proving a version of the theorem for primes in arithmetic progressions with uniformity in the modulus, as well as a non-abelian analogue of this.

On a conjecture of B. C. Kellner

The aim of this note is a proof of a recent conjecture of Kellner concerning the number of distinct prime factors of a particular product of primes. The proof uses profound results from analytic number theory, such as Granville-Ramaré's estimate of an exponential sum over primes.

Combinatorial Identities Involving Mertens Function Through Relatively Prime Subsets

In this note we give some identities which involve the Mertens function M(n). Our proofs are combinatorial with relatively prime subsets as a main tool.

素数数字分布猜想：素数中平均数字等分布的形式证明

The Prime Digit Distribution Conjecture: A Formal Proof of Average Digit Equidistribution in the Prime Numbers

研究素数数字分布，结合素数定理等方法证明素数中数字出现概率渐近为\(\frac{1}{10}\)，建立内部数字定量等分布，阐明与其他相关问题区别，得出素数数字汇总后渐近等分布的结论。

On Mullin's second sequence of primes

We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.

On the difference between consecutive primes

Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new results. We show that the sum of squares of differences between consecutive primes $\sum_{p_n\le x}(p_{n+1}-p_n)^2$ is bounded by $x^{5/4+ε}$ for $x$ sufficiently large and any fixed $ε>0$. The proof relies on utilising various mean-value estimates for Dirichlet polynomials.

关于算术级数中素数的Linnik估计的一个pretentious证明

A pretentious proof of Linnik's estimate for primes in arithmetic progressions

本文采用pretentious方法，基于pretentious大筛法思想与积性函数处理思路，证明了算术级数上von Mangoldt函数之和的强一致估计，该估计与Linnik的相关经典估计类似。

Variations of Lucas' Theorem Modulo Prime Powers

Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem modulo prime powers holds $$ {np^k +n_0 \choose mp^k+m_0}\equiv{np^{\lfloor(k-1)/3\rfloor} \choose mp^{\lfloor(k-1)/3\rfloor}} {n_0 \choose m_0} \pmod{p^k}. $$ In the proof the authors used their earlier result that present a generalized version of Lucas' Theorem. In this paper we present a a simple inductive proof of the above congruence. Our proof is based o…

On the Primes in the Interval [3n, 4n]

For the old question whether there is always a prime in the interval [kn, (k+1)n] or not, the famous Bertrand's postulate gave an affirmative answer for k=1. It was first proved by P.L. Chebyshev in 1850, and an elegant elementary proof was given by P. Erdos in 1932. M. El Bachraoui used elementary techniques to prove the case k=2 in 2006. This paper gives a proof of the case k=3, again without using the prime number theorem or any deep analytic result. In addition we give a lower bound for the number of primes in the interval [3n, 4n], which shows that as n tends to infinity, the number of pr…