For questions on prime twins.

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1answer
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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 ...
4
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0answers
137 views

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 ...
0
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0answers
46 views

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 ...
2
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0answers
29 views

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 ...
3
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1answer
67 views

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 ...
0
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1answer
43 views

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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0answers
35 views

Question about twin primes

How do I understand the last statement of the hint..?
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0answers
73 views

Can I use integer frequencies in quadratic intervals to set a lower bound for primes? [closed]

I want to find out if the following arithmetic approach could produce a backdoor proof of Legendre’s Conjecture. There are two assumptions, Questions A and B, which are posed in the text and labeled ...
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2answers
341 views

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}, ...
5
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3answers
213 views

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 ...
3
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0answers
48 views

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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0answers
17 views

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 ...
7
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2answers
165 views

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 ...
0
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2answers
61 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 ...
4
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1answer
71 views

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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0answers
55 views

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 ...
7
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3answers
2k views

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 ...
2
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1answer
154 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 ...
31
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3answers
7k views

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 ...
0
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0answers
51 views

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., ...
2
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1answer
92 views

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 ...
2
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0answers
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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 ...
4
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5answers
860 views

Determining the next Twin Prime?

A really simple I question I guess. Is there an algorithm or method such that given an integer $N$ there is a way to determine the next twin prime pair greater than $N$? If yes, then could you please ...
10
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1answer
286 views

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 ...
4
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1answer
77 views

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 ...
11
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3answers
1k views

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 ...
8
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2answers
483 views

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 ...
0
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1answer
76 views

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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0answers
35 views

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
292 views

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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1answer
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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 ...
85
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15answers
8k views

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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72 views

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
185 views

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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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
188 views

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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0answers
61 views

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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0answers
43 views

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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2answers
83 views

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
39 views

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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3answers
399 views

Twin primes satisfy the congruence?

I need a justification for my observation. In general, we can list twin prime pairs in $(6n-1, 6n+1)$, where $n$ is some positive number. Of course, this is valid except $(3, 5)$. Now, I construct, ...
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0answers
68 views

Lower bound for twin prime counting function

We know an upper bound for twin prime counting function. If i'm not mistaken it was (x*loglogx)/(logx*logx). Do we have a lower bound for it? Is it the same function? I mean is there c constant such ...
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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
237 views

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...)
4
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1answer
322 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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3answers
131 views

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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1answer
47 views

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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0answers
114 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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0answers
95 views

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 ...
7
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0answers
106 views

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 ...