For questions on prime twins.

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Is there a good database with math papers?

I'm trying to do extensive research on the Twin Prime conjecture for one of my undergraduate classes, but I'm having trouble finding the sources. For example, I can only find Polignac's paper in ...
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Step to prove twin primes' conjecture: $\liminf_{n\to\infty}(p_{n+1}-p_n)<7\cdot10^7$

Today I have found that the Chinese mathematician Yitang Zhang has proven in 2013 that the sequence $d_n=p_{n+1}-p_n$ where $p_n$ is the $n$th prime has a finite inferior limit (and in fact, lesser ...
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Prove that there exists infinitely many primes of Digital root $2,5$ or $8$

I am highly interested in properties of digital root. Digital Root: Digital root of a number is a digit obtained by adding digits of number till a single digit is obtained. It's clear that Digital ...
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1answer
39 views

Proving the primality of these large numbers?

In 2007, Vautier claimed that the largest known consecutive pair of prime numbers (at the time) was $2003663613\cdot2^{195000}-1$ and $2003663613\cdot2^{195000}+1$. I was wondering how Vautier found ...
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185 views

Is this statement equivalent to Goldbach's conjecture

Given a number $n\ge 3$, then one of these is true: \begin{equation} \begin{cases}2n = (6m-1)+P, \ \ \ P \in \mathbb P, \ 6m-1 \in \mathbb P, \ 6m+1 \in \mathbb P \ \ \ \ (1) \\ 2n-1 \in \mathbb P, \ ...
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Twin prime conjecture hypothesis

Let $c$ be a positive integer and fix $a=c-1$, and $b=c+1$. Challenge: Find the largest value of $c$ such that $ac\pm1$ and $bc\pm1$ are pairs of twin primes. For example, with $c=6$ we have ...
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On an exercise from a journal using Hölder and Stoltz theorems, now with twin primes

I use [1] (in spanish) for the sequence of positive terms defined by $$ a_k = \begin{cases} \frac{1}{k}(\frac{1}{p_k}+\frac{1}{p_k+2}), & \text{for the kth twin prime pair} \\ 0, & \text{if ...
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105 views

Prime-twins and infinite products

For $n\geq 1$ let the nth twin prime pair $$(p_n,p_n+2).$$ This sequence start as $(3,5),(5,7)$, the next $(11,13)\ldots$. I have two short questions about twin primes and infinite product defined ...
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Cousin primes in the Ulam Spiral

I was plotting the Ulam spiral (https://en.wikipedia.org/wiki/Ulam_spiral), and decided to isolate twin/cousin/sexy primes on the Ulam spiral. Although plotting twin primes offered no obvious lines, ...
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Would this sequence (OEIS A068374) be somehow attached to the twin prime conjecture?

Today I came across an interesting sequence at OEIS, A068374, described as "Primes $n$ such that positive values of $n$-Primorial($k$) are all primes ($k\gt0$)". The sequence is as follows: $(2, ...
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A conditional asymptotic for $\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$, when $\alpha>-1$

When I've followed a notes that show how obtain a similar asymptotic using Abel summation formula, my case with $a_n=\chi(n)$, the characteristic function taking the value 1 if $p$ is prime (in a twin ...
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41 views

Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove ...
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Twin primes sums conjecture

I have found an interesting conjecture between twin primes sums. I don't know if it is already described by someone else. I have checked in internet, but I didn't find any mention of such conjecture. ...
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What's that work that shows there are infinitely many $2k$-prime pairs for some large enough $k$?

There was something published that said there are infinitely many pairs of primes that differ by $2k$ for some large $k$. Can you help me find it? Thanks.
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Modified Euler's Totient function for counting constellations in reduced residue systems

I am working on a modified totient function for counting constellations in reduced residue systems for the same range that Euler's totient function is defined over. This post is separated into three ...
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1answer
56 views

Which heuristic leads to the Hardy-Littlewood conjecture about twin primes?

According to Wikpedia, Hardy-Littlewood conjecture says that $$\pi_2(n) \sim 2 C_2 \frac{n}{(\ln n)^2} \sim 2 C_2 \int_2^n {dt \over (\ln t)^2}$$ where $$C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} ...
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Sets of Prime and Composite Numbers

We know that all primes are of the form $ 6k ± 1 $ with the exception of 2 and 3. We also know that not all numbers of the form $ 6k ± 1 $ are prime. This leads to four distinct sets of pairs ...
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Twin square-free numbers of the form $6k-1,6k+1$?

Is it easy to show (or even known) that there are infinitely many square-free pairs $6k-1,6k+1$? (Presumably, not disproven yet, since a lot of people would be wasting their time on the twin prime ...
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How do we identify twin primes .

as known , each prime number greater than 3 is of the form $6k-1$ or $6k+1$ . twin primes are all sort of two adjacent primes of difference $= 2$ as: $$(11,13) ,(17,19),\ldots,(6k-1,6k+1)$$ -Is ...
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For which $k$ with $0<k<210637$ is $k\times 3571\# \pm 1$ a twin-prime-pair?

Because PARI/GP is not very fast in primilaty testing, I did not check the pairs $k \times 3571\# \pm 1$ in ascending order, but I begun with $k=200,000$ and got the twin prime pair $$210637\times ...
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Is it known whether $6\times 10^n\pm 1$ is a twin prime for some $n>2$?

I checked the number pairs $6 \times 10^n \pm 1$ for $1 \le n \le 2000$. The only twin primes of the desired form I found are: $(59, 61)$ and $(599, 601)$. I wonder if these are the only pairs. ...
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Why doesn't this twin prime counting function work?

Quite some time ago, I made a function $f(x)$ which I thought would give me the minimum amount of prime twins equal to or lower than $x$. I have tested this function for large values of $x$ and it ...
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Is a probable prime known larger than the largest known prime?

According to Wikipedia, the largest known prime is $2^{57,885,161}-1$ with $17,425,170$ digits. Because a probable prime is usually easier to find than a proven prime (although for the ...
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130 views

Elementary Twin Prime Attempt. [closed]

There's a theorem somewhere that for sufficiently large $k$ there exists an infinite number of prime pairs with difference $2k$. Let $\ell$-prime pair mean a pair of primes separated by a distance of ...
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How to pigeonhole the primes between $p_n$ and $p_{n+1}^2$ for twin prime conjecture?

For any full list of the primes up to the $n$th prime: $P = \{2, 3,5,\dots, p_n\}$, any natural number $q$ such that $ p_n \lt q \lt p_{n+1}^2$ that is not sieved by a prime in $P$ is also a prime. ...
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160 views

Does the Riemann-Hypothesis imply the Twin-Prime-Conjecture?

The Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis) is one of the most important conjectures in number theory. I read that the Riemann hypothesis implies the Goldbach Conjecture ...
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56 views

Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?

Could someone shed some light on what we know about the density of twin primes? I find that it seems to be empirically true that the density of prime gaps increases as $\log(x)$ does for any gap. ...
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Polynomial equations in $p$ and $q$ with $p,q$ primes

Is there a non zero polynomial $R \in \mathbb{Z}[X,Y]$ such that there exists an infinite number of pair $(p,q)$ with $p$ and $q$ primes, $p \neq q$ and $R(p,q)=0$ ? I know the curve must be of ...
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Is this a valid equivalent expression of the twin prime conjecture?

The twin prime conjecture states basically that it is possible to find two primes $p$, $p+2$ at a distance $2$ that are as big as wanted (Wikipedia). I am learning about the basic properties ...
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inequality on Gaussian prime

Consider any Gaussian prime $p$ (except $|p|=\sqrt{2}$). If we have $|x|\leq|p|+0.5$, where $x$ is a nonzero Gaussian integer, can we prove $|x|\leq|p|$?
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A sum involving twin primes and Prime Number Theorem

This morning I've been watching documentary about asterorids, in a scene an astronomer explains the so called image subtraction process or pixel subtraction, a mathematical model used in computerized ...
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How is the Twin Primes Constant useful? What value does it provide over Brun's Constant?

The Twin Primes Constant is: $$\prod_{p > 2 \text{ and a prime }}\left(1 - \frac{1}{(p-1)^2}\right) = 0.6601618158\ldots$$ It appears that in this case $p$ does not have to be a prime. But if ...
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58 views

Furstenberg theorem and twin primes

The theorem of Furstenberg showing there exists infinitely many primes (and variants, including those stripping away the topological side of things) has been discussed several times on MSE, e.g. in ...
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Estimating total number of twin primes

Taking my notation from a previous question Define a function $P_6$ as $$P_6(n)=\begin{cases} 0, \ \ 6n-1 \not\in \mathbb P \wedge 6n+1 \not\in \mathbb P \\ 1, \ \ (6n-1 \not\in \mathbb P \wedge ...
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135 views

Looking for help to clearly define a function that counts the number of twin primes in a range

My goal is to define a function that counts the number of twin primes between $q$ and $q^2$ where $q$ is any prime greater than $7$. I would like to do this using: The Sieve of Eratosthenes The ...
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About the total number of twin primes in the vicinity of twin primes

Just for curiosity's sake, I did a test regarding twin primes, and I have doubts about the meaning of the results. Test: calculation of ${\pi_2}$(n) and the twin primes density in the vicinity of ...
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Looking for a function which can serve as an upper bound to a count of the the pairs (x)(x+2) that have a given least prime factor?

Let $p \ge 7$ be a prime. Let $z > p$ also be a prime. Let $f_p(z)$ be the number of elements $x$ such that $z \le x < z^2$ and the least prime factor of $x(x+2) = p$ I am trying to find ...
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If $q$ is a prime, $gcd(x(x+2),q\#)=1$ and $q < x < q^2$, doesn't it follow that $x,x+2$ are twin primes?

I recently asked a question that was not well received. That's ok. I don't disagree with the ratings if my question is unclear. I want to verify the foundation of my reasoning. Doesn't it follow ...
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Distribution of composite numbers

This question is moved from mathoverflow, there are several excellent answers at mathoverflow which improve my question greatly. For more information, please see the original question posted on ...
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Form of $k$ when both $6k-1$ and $6k+1$ are primes

After a quick glance at sequence A007693 it seems that the following is true: if $p$ and $p+2$ are prime, then $\frac{p+1}{6}$ is prime. Questions: a) Is it the case? If not, what is the ...
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Aren't there obvious patterns in the primes that no one makes use of and what about this…

Let's take the sequence of naturals at or above two ($2, 3, 4, \dotsc$) and cross out just the primes $2$ and $3$, as well as all their multiples: $$\require{cancel}\cancel{2}, \cancel{3}, \cancel{4}, ...
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Are there any known special properties of a number located between twin primes?

With the exception of $4$, every number located between twin primes is divisible by $6$. This one is obvious, but are there any other properties that can be ascribed to such numbers? A property may ...
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Positive proportion sets of numbers not divisible by twin primes.

Is it possible to explicitly construct a set of integers $S$ which contains a positive proportion of the positive integers and every integer in $S$ is not divisible by any prime $p$ in the set of ...
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What percentage of numbers is divisible by the set of twin primes?

What percentage of numbers is divisible by the set of twin primes $\{3,5,7,11,13,17,19,29,31\dots\}$ as $N\rightarrow \infty?$ Clarification Taking the first twin prime and creating a set out of its ...
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Is there a link between the Bunyakovsky conjecture and the Twin Prime conjecture?

Can the proof of one conjecture be considered a proof of the other conjecture? The general method of building an infinite number of prime producing quadratic polynomials was given in the link ...
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Are there infinitely many Thâbit ibn Kurrah cousin primes?

Positive integers of the form $3 * 2^n - 1$ are called Thâbit ibn Kurrah numbers. and if those numbers are prime they are called Thâbit ibn Kurrah primes. Now if for a fixed positive integer $n$ , ...
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Prime twins and $1 \mod 30$ confusion

Jie Wu improved Brun's theorem and showed that the number of prime twins up to $n$ satisfies for sufficiently large $n$ : $$\pi_2(n) < 4.5 \frac{n}{ln(n)^2} $$ However this confused me while ...
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What would be the immediate implications of a formula for prime numbers?

What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
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185 views

At what point does the number twin prime between $n^2$ and $(n+1)^2$ stop increasing in count?

This question was so well stated by someone else that I am quoting their words here: Let $a(n)$ be the number of pairs of twin primes between $n^2$ and $(n+1)^2$. Of course, if the twin primes ...
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Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who ...