For questions on prime twins.

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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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1answer
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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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1answer
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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 ...
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2answers
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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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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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1answer
128 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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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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1answer
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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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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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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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2answers
66 views

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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4answers
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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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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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1answer
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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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1answer
162 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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The event: ($p$ is a prime and $p+2$ is also a prime) occurs *independentely* from the previous couples of twin primes: $(q,q+2)$ where $q<p$.

I am asking if there is a mathematical proof that the event: ($p$ is a prime and $p+2$ is also a prime) occurs independently from the previous couples of twin primes: $(q,q+2)$ where $q<p$, i.e., ...
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Reasoning about prime counting and the twin prime conjecture

I've been thinking the primorial for say the $i$th prime $p_i$and the equations for counting the number of elements in the reduced residue system for this primorial and counting the number of elements ...
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First Order Logic: Prove that the infinitely many twin primes conjecture is equivalent to existence of infinite primes

I'm learning First Order Logic independently using a college textbook. I've been doing some self exercise question in it and came across this one, which I can't seem to figure out how to do: Let ...
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1answer
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Twin Primes between $n$ and $2n$

Is it theoretically possible for there to always be a twin prime pair between $n$ and $2n$ for all sufficiently large $n$ (assuming of course that there are infinitely many twin primes) or would this ...
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What is wrong with this proposed proof of the twin prime conjecture?

I was thinking on the twin prime conjecture, that there are an infinite number of twin primes... I came up with a proof. I have to think that it is incomplete or wrong, because many great minds ...
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Is there is any result claiming that there cannot be any other twin Mersenne primes?

There are 3 known Twin Mersenne Primes: $M3$ and $M5$, $M5$ and $M7$, $M17$ and $M19$. More precisely, if both $M(p)$ and $M(p+2)$ are both prime, then they are called Twin Mersenne Primes. My ...
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1answer
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Let $g_n^k=p_{n+k}-p_n$, where $p_n$ is the $n$th prime. Does there exist $g_{k+1}^1=2$ such that $g_1^k,g_2^k,\ldots$ is a “Gilbreath sequence?”

Call $(S_i)_{i=1}^{\infty}$ a Gilbreath sequence if $1=\lvert S_2-S_1\rvert=\lvert \lvert S_3-S_2\rvert-\lvert S_2-S_1\rvert\rvert=\cdots$, i.e., if the sequence can be substituted for the primes in ...
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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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Asymptotic Expression for the Twin Prime Counting Function

A variation on a previous question I asked, which has garnered no responses. I'll attempt to be more lucid: Let $\pi_2(x)$ be the twin prime counting function and $\pi(x)$ be the prime counting ...
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Prime Reflections

How would you describe the following pattern?: For each primorial from 30 onward, there exists a pattern in the arrangement of the prime factors of the composite numbers which I call "the mirror ...
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2answers
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What if a conjecture were provably unprovable?

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false. Is it then possible for the conjecture to be true? ...
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1answer
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Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly ...
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Can we abstractly construct this ring and what is it isomorphic to?

Define $R = \Bbb{Z}[X_1, X_2, \dots]$. Then place on $R$ the relations $$ X_1 + 1 = X_2, \\ X_2 + 2 = X_3, \\ X_3 + 2 = X_4, \\ X_5 + 2 = X_6, \dots \\ X_{2k-1} + 2 = X_{2k}, \ \forall k \geq 3 ...
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2answers
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About twin primes and their happy mothers.

Let's say that a positive integer $n$ is a happy mother if $6$ divides $n$ and $(n-1,n+1)$ is a pair of twin primes. Is the difference between two consecutive happy mothers necessarily a happy mother ...
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About a paper by Gold & Tucker (characterizing twin primes)

I've carefully looked at the questions on prime and twin prime, but the following question seems not to habe been asked before. Context: In the paper by Jeffrey F. Gold and Don H. Tucker titled A ...
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Estimating the Twin prime constant

http://numbers.computation.free.fr/Constants/Primes/twin.html it says: "This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be ...
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Conjecture on twin primes

Let $p$ and $p+2$ be both prime. I conjectured (with my ignorance) that $$p^{\frac{p+1}{2}}\equiv -1\mod{(p+2)}$$ except for $p=17,41,71,137, 191, 239....$ I verified this on Mathematica. So for ...
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1answer
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Is there a generalization of the twin prime conjecture to rings or certain rings?

The question's in the title. For instance, if $R$ contains $2$ then there are an infinite number of pairs of prime principal ideals $(p),(q)$ such that $p = q + 2$. I just made that up and it's ...
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1answer
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About Brun's Theorem

http://arxiv.org/pdf/1401.7555.pdf On the page 8 there is a proof of Brun's theorem. $$\large-\int_1^{\infty}\pi_2(x)\;\mathrm{d}\left(\frac1{\lfloor x \rfloor}\right)=-\sum_{n\ge ...
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2answers
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Twin Prime Conjecture's Proof [closed]

I've found this article that claims to have a proof of the Twin Prime Conjecture. Can you find any error? (I have some doubts about the last page of the paper...)
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1answer
355 views

Can the twin prime conjecture be solved in this way?

After some research, I have discovered that proving the statement; There exist an infinite number of positive integers K such that; $K \neq 6ab \pm a \pm b$ and $K \neq 6ab \mp a \pm b$ is ...
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1answer
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Infinite “Twins” in reduced residue systems modulo primorials

The Lth primorial ($p_L\#$) is the product of the first L prime numbers. The reduced residue system modulo $p_L\#$ is any set of positive integers with cardinality equal to the totient of $p_L\#$ ...
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Bounds on the Twin Prime counting function

Maybe I'm mistaken, but wouldn't it suffice to show that the twin prime counting function has any kind of strictly increasing lower bound to show that there are infinitely many twin primes?
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Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are ...
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1answer
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Can we determine which statements are incomplete due to Godel?

Due to Godel's incompleteness theorems we know that there are true statements in a system that cannot be proven with that system. My questions are 1) can we tell which statements in a system are the ...
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1answer
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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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1answer
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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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For primes $P_1$ and $P_2$, exists a prime $P_3$ that both $P_i + 6P_3$ is a prime

I was thinking about twin primes and I came to ask this question: If we have two distinct primes $P_1$ and $P_2$ which are both greater than $3$, then does there always exist a prime $P_3$ such that ...
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2answers
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Twin prime “test” via congruence

I decided to try getting a test for a "twinness" of a prime via Wilson's theorem. Wilson's theorem says that integer $n > 1$ is a prime iff $$(n-1)! \ \equiv -1 \pmod n $$ Now, if both $n$ and ...
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1answer
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Let $p_k$ be the $k$th prime, can it be shown for $p \ge 5$, that there is not always a twin prime between $p_k^2$ and $p_{k+1}^2$?

For any primorial $p_k \ge 3$, $p_k\#$, there are $$\prod_{2\le{i}\le{k}} (p_i-2)$$ distinct instances of $x,x+2$ that are relatively prime to $p_k\#$. If any of these pairs are less than ...
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2answers
109 views

Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?

Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?