For questions on prime twins.

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What's about of an analogous Riemann's function $R(X)$ for twin primes?

It is well know the so-called Riemann's explicit formula for the prime counting function $\pi(x)$ involving the density $J(x)$ for prime powers and how by Möbius inversion one recovers $\pi(x)$ and ...
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Primes that are neither twin, cousin or sexy [closed]

I'm reading up on prime pairs, and I had a question... I can't seem to find an answer to this anywhere, and the wikipedia list of prime types is enormous! Afraid I missed it when going through it. I ...
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Why can the sieve of eratosthenes not be used to confirm the twin primes conjecture?

I have been having fun thinking about sieves and more particularly the twin prime conjecture. As I am fairly new to this type of mathematics, I am wondering, if we use the sieve of erastothenes, aka ...
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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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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 ...