For questions on prime twins.

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Characterization of primes $(6n+1, 6n-1)$ that are not twins

According to OEIS Sequence A002822(https://oeis.org/A002822), it states that $6n+1$ is a twin prime $iff$ $n$ is not of the form $6ab \pm a \pm b$. I was wondering if anyone had a proof for this. ...
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Confusion about Saibians article about primes [on hold]

Here : https://sites.google.com/site/largenumbers/home/1-5/2 Saibian claims that it was proven that infinite many twin primes exist. Did I miss something ? Isn't the twin-prime-conjecture open ?...
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Empty Twin Prime Sets

Consider this set of numbers: $1, 5, 8, 11, 13, 31, 37, 53, 61, 73, 79, 97, 122, 127$ This is the set of numbers $n$ such that $nm \pm 1$ is not a twin prime pair for all $m \leq n$. For instance, $...
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Are they twin primes?

Let $n$ be a positive integer and let $p_1, p_2,...,p_n$ be first n primes. And let $m$ be the smallest integer $m\ge\,p_n$ such that $m$ and $m+2$ are coprime to $p_1,p_2,..., p_n$. Are $m$ and $m+2$...
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Asymptotic density of Zhang's primes

By this point, it is well known that Yitang Zhang's result implies for some $c$, there are infinitely many primes $p$ such that $p+c$ is also prime, and that the smallest such $c$ is less than $70,000,...
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for a chen prime p, what is the size of factors of p+2

Suppose the twin prime conjecture fails. Then, by Chen's theorem, there are infinitely many primes $p$ s. t. $p+2$ is a product of exactly two primes. It would be nice to know that as $p$ grows, so ...
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Factorial and primorial twin primes

Factorial primes are are primes of the form $n! \pm 1$ and primorial primes are primes of the form $p\#\pm 1$, where $p\#$ is the product of all primes $\leq p$. To cite http://www.ams.org/journals/...
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How can I calculate OEIS A144311 efficiently?

I'm looking for a way to calculate OEIS A144311 efficiently. In one sense or another, this series considers the number between "relative" twin primes. What do I mean by this? Well, the number $77$ ...
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Is this a new twin prime sieve method? Any information or comments is very appreciated.

I'm studying the twin prime numbers. Instead of sieving prime numbers, I found this method to sieve $\{x: x \neq \pm 1 \text{( mod $p$)}, x \in \mathbb{N}, p \le p_i\}$, so that $(x-1,x+1)$ will be ...
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Can you share some information to help study this unified sieve function for prime, twin prime and Goldbach sums of $2n$?

Let $p_i$ be the $i^{th}$ prime number. For Goldbach sums of $2n$, let $p_i$ be the largest prime less than $\sqrt{2n}$, define $$ P(p_i,n,x)=\sum_{p\le{p_i}}\frac{c_p}{p}\left(1+2\sum_{k=1}^{p-1}(1-\...
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Consecutive Answers to Chinese Remainder Theorem

We'll start with 2 congruences only. We'll allow only numbers that when divided by 6, don't have a remainder of 3. Also, only numbers that have a remainder of 2 or 4 when divided by 5: $\equiv ${$0,1,...
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Twin Prime Max Gaps

Ok, let's build a foundation here: A common way of testing primality, is dividing by all primes smaller than the number's square root. For instance, $97$ is prime because dividing by none of the ...
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Twin prime gaps

I realize that because it is only conjectured that there are infinitely many twin primes, we can't say, "There will always be a twin prime between _____ and _____." Like we can for primes. Still, do ...
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Recreational math dealing with twin primes

This is kinda recreational math with a goal in mind of progressing further toward a proof of the twin prime conjecture. Consider this: We start with a random prime: $109$ $3*109=327$ $327 \equiv$...
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Is this proof of the twin prime conjecture? [closed]

Identifying twin primes [1] Any natural number $n : 1<n\leq p_x^2 $ where $n$ is not divisible by any prime number less than $p_x$ is a prime number, except when $n$ is one of those prime ...
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Twin Primes (continued research)

This has become increasingly crowded, so at the onset, let me state this: My question is, is there some reason this is so linear that I'm not seeing? The only thing it seems to indicate to me is that ...
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Twin Primes, their Arithmetic Means and some properties.

These are two problems which I have been trying to solve. The arithmetic mean of twin primes 5 and 7 is 6 which is a triangular number. Do there exist any other such twin primes? If they exist ...
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Source for Proof of Brun's Theorem?

I posted previously asking for advice on a paper I'm writing for a senior math class, and I have made a lot of progress. I have come to the point where I want to prove Brun's theorem and talk about ...
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Proof involving twin primes

I have to prove that if $p$ and $p+2$ are twin primes, $p>3$, then $6\ |\ (p+1)$. I figure that any prime number greater than 3 is odd, and therefore $p+1$ is definitely even, therefore $2\ |\ (p+...
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Twin Prime Related Material

I'm in a senior seminar class for my undergraduate degree and I am tasked with writing a short, 12 page paper on some subject I have not been taught before. I chose the twin prime conjecture. My ...
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Would this be an easier way to prove the twin prime conjecture?

Prove: For every prime, $p\geq7$, there exists some $pn$ such that $p$ is its largest prime factor, $n$ is a positive integer, and $(pn-4, pn-2)$ is a twin prime. My questions: Would this indeed ...
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Algebraic identity involving powers of twin primes

Yesterday, I verified that, if $a$,$b$ and $c$ are real numbers such that $a+b+c=0$, then $$\frac{a^5+b^5+c^5}{5}=\frac{a^3+b^3+c^3}{3}\cdot\frac{a^2+b^2+c^2}{2}$$ and $$\frac{a^7+b^7+c^7}{7}=\frac{...
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Is there more than one occurrence of a power of two between twin primes?

$2^2$ is between the twin primes $3$ and $5$. Are there any other instances of a power of two between twin primes? If so, how many? That there are Mersenne primes (primes of the form $2^n-1$) ...
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Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that $$\mathcal{B}=\sum_{n\geq 1}\left(\frac{1}{...
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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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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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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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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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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 $a=...
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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, 5, ...
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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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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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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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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 Mersenne-...
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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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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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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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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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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 ...