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Questions tagged [prime-twins]

For questions on prime twins.

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How do you compute the singular series?

Terence Tao gives at his blog the following formula for something called the singular series: $$\large\mathfrak{S}(h)=2\Pi_{2}\prod\limits_{p|h;p>2}\frac{p-2}{p-1}$$ where $\Pi_{2}=0.66016...$ is ...
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The significance and acceptance of Helfgott’s proof of the weak Goldbach Conjecture

Recently I was browsing math Wikipedia, and found that Harald Helfgott announced the complete proof of the weak Goldbach Conjecture in 2013, a proof which has been accepted widely by the math ...
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Doesn't the first Hardy-Littlewood conjecture imply the finiteness of prime constellations?

The first Hardy-Littlewood conjecture says, in essence, that if all numbers within a prime $k$-tuple do not form a complete residue class with respect to any prime, then they are infinite in number ...
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Why aren't $P-1,P+1$ primes where $P = p_1p_2…p_n$? [duplicate]

Euclid's theorem states: Consider any finite list of prime numbers $p_1, p_2, ..., p_n$. It will be shown that at least one additional prime number not in this list exists. Let $P$ be the product ...
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Iterated Twin Prime conjecture

Here is the beginning of the list of sums of twin prime pairs (OEIS A054735): 8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924,... "...
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Proving the divisibility of $4[(n-1)!+1]+n$ by $n(n+2)$ in the condition of $n,n+2 \in P$ where $P$ is the set of prime numbers [duplicate]

Let $n$ and ($n+2$) be two prime numbers. If any real value of $n$ satisfies that condition, then prove that $$\frac{4{[(n-1)!+1]}+n}{n(n+2)} = k$$ where $k$ is a positive integer. SOURCE: BANGLADESH ...
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Finding primes from 6 integers closest to two twin primes multiplied together.

We are given the twin primes $a$ and $b$, where $a > 5$. We are told that only one of the following: $ab-3, ab-2, ab-1, ab+1, ab+2, ab+3$ will sometimes generate a prime but not always. It's ...
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Is there a counter-example to these number theoretic conjectures?

Question and Summary I recently made the following heuristic observations: Let, $$ xy = p_1^{a_1} p_2^{a_2} \dots p_n^{a_n} $$ where $a_i\geq1$ Conjecture $1$: then there must exist $x-y=p_{n+1}$ ...
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What is the largest known twin-prime of the form $2^a\cdot 3^b\pm 1$?

$$2^{2176}\cdot 3^{2175}\pm1$$ is an example of a twin-prime pair of the form $$2^a\cdot 3^b\pm 1$$ with positive integers $a,b$. Each prime "only" has $\ 1\ 693\ $ digits, so larger examples probably ...
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How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$? [closed]

How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$? Examples, for $n=3$, there is a twin prime between 9 and 25 of (11,13). For $n=9$, there is a ...
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1answer
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Methods for proving a function outputs an infinite number of integers

I have a function involving polynomials and the centre of the Binomial Triangle and I'd like to prove that the function produces a positive integer infinitely many times. I don't have any interest in ...
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1answer
91 views

Improved sieve for primes and prime twins?

Suppose we want to estimate the number of primes between $x$ and its square root, say for example between $10$ and $100$ with a sieve. There are $90 $ numbers so we estimate : $\pi(10,100) = 90(1-1/...
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1answer
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Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs?

Is it true for any $n=2p$ where $p$ is prime, that the number of twin primes less than $n$ approaches the number of prime pairs $(p_{1},p_{2})$ such that $p_{1} + p_{2} = n$? For example, If we ...
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How accurately must I compute the twin prime constant to get the twin prime density?

Let $\pi _{2}(x)$ denote the number of primes $p\leq x$ such that $p+2$ is also prime. Hardy and Littlewood conjectured that $$ {\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}...
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Prime numbers falling in the gap between twice the members of twin primes

Frequently, the gap between twice a pair of twin primes contains a prime number. That is, for $p_i,(p_i+2)\in \mathbb P$, it is often but not always the case that one of $2p_i+1$ or $2p_i+3$ is also ...
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1answer
60 views

Twin prime conjecture and gaps between primes

This is just a thought: if gaps between prime numbers can be arbitrarily large then it should be possible to find infinitely many gaps, such that the product $m=\prod_{n=1}^{N}Pn<P_{N+1}^{2}$, ...
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1answer
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On a theorem of Zumkeller related to twin primes

Let $\{p_1,p_2\}$ be a twin prime pair, $\phi(n)$ denote Euler's totient function and $\sigma(n)$ the sum-of-divisors function. Reinhard Zumkeller proved in 2002 that $$ \phi(p_2) = \sigma(p_1). $$ ...
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A conjecture on the closeness of twin primes

Let $p_1$ and $p_2$ be twin primes, and let $p_1-1=a_1\times b_1$ and $p_2+1=a_2\times b_2$ be such that $|b_1-a_1|$ and $|b_2-a_2|$ are minimised. Similarly, let $p_1+1=p_2-1=a\times b$ be such that ...
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A problem inspired by the Twin Prime conjecture

I came up with a question several hours ago...but I couldn't find any information about it. The problem goes like below $$P^n_k =\{(p_1,...,p_n)|p_1<...<p_n:primes,p_n-p_1\le k\}$$ $$k_n=min\{...
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1answer
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Even numbers of the form $\frac {n(n+1)}{2}$ and twin primes

If we take some even number of the form $\frac{n(n+1)}{2}$ and add $1$ to it and also subtract $1$ from it then we have a mapping $\frac{n(n+1)}{2}\to\left\{\frac{n(n+1)}{2}-1,\frac{n(n+1)}{2}+1\right\...
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Prime twins and $ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $

Let $p$ be a prime such that $p+2$ is Also a prime. Define $$ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $$ For $z \neq 1 $ and $re(z) > 1$ this $f(z)$ always ...
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Consequences of finite many twin primes?

Suppose , it turns out that the number of twin primes is finite (this is very unlikely, but let us assume it). Which consequences would such a result have for number theory ? To be more concrete :...
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Can't find the error in this proof of the twin prime conjecture

Just to be clear: I don't think that this proof is valid; both it and I are far to simple to have proven the twin prime conjecture. I am only unable to find any mistakes in it. Let $a$ be any prime ...
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The sieve formula choosen in Zhang's breakthrough work in Twin Prime conjecture

In the breakthrough work of the proof of weak twin prime conjecture, Goldstone, Pintz and Yildirim as well as Zhang use the following modified Selberg sieve: $v=\lambda^2$ where $\lambda(n)$takes ...
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1answer
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Is there public access to a paper, in which the first $k$-tuple conjecture was proposed? [closed]

How did Hardy and Littlewood derive this conjecture and what needs to be done to prove it?
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A twin prime theorem, and a reformulation of the twin prime conjecture

In a previously posted question (A sieve for twin primes; does it imply there are infinite many twin primes?), I demonstrated that a sieve can be constructed that identifies all twin primes, and only ...
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1answer
175 views

A sieve for twin primes; does it imply there are infinite many twin primes?

I have devised a sieve for identifying twin primes. My first question will be: Have I just rediscovered something already known? By comparing my sieve to the Sieve of Erastosthenes, I argue that there ...
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1answer
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Probability of attaining same prime factor [closed]

EDITED: Two coloums of numbers are presented, there are 234 entries in both columns. Column A, values are not multiples of 19, natural numbers increasing by a random value with mean of 30. Column B, ...
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Is it true that every pair of twin primes >3 is of the form 6n plus or minus 1? [closed]

Is it possible to have twin primes whose center is not divisible by 6?
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Brun’s constant and irrational numbers

It is trivial that if there are finitely many twin primes then Brun’s constant must be a rational number. And GammaTester (below) has offered an example of an infinite series that converges to a ...
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Twin prime pairs of the form $(\prod_{j=1}^n \phi(j))\pm 1$

Let $$f(n):=\prod_{j=1}^n \phi(j)$$ where $\phi(n)$ denotes the totient-function. I searched for twin primes of the form $f(n)\pm 1$. Upto $n=3\ 000$, the $n's$ I found are $[4, 7, 12]$. What is ...
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Twin indices of the form $K = 6 (a n)^b$

Let $K \ge 6$ (usually called twin index) be the number between a pair of twin primes, and let $k = K / 6$. It is easy to see that all $k = n^2$ (where $n$ is a generic integer) are divisible by ...
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4answers
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Twin primes of the form $n^2+1$ and $N^2+3$?

Assume that there are infinity many primes of the form $n^2+1$ and there are infinity many primes of the form $N^2+3$ , Then could we show that there are infinity primes of the form $n^2+1$ and $...
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1answer
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A symbolic combination of an inequality concerning convergent series and Brun's theorem: improving the upper bound

We define from the sequence of twin primes, see this MathWorld the following sequence $$t_n:=\sum_{\substack{1\leq k\leq n\\p_k\in\mathcal{T}}}p_k\tag{1}$$ where we denote the set of all twin primes ...
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How often is the number between two twin primes divided by 6 a prime?

This question has been edited thanks to the feedback by one user: 12 is in between 11 and 13, and 12/6 = 2 which is prime. So if we take 29 and 31, 30 is in between, and 30/6=5 which is prime In ...
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Can every multiple of $3$ be written as arithmetic mean of two pairs of twin prime numbers ???

Can every multiple of $3$ be written as arithmetic mean of two pairs of twin prime numbers ??? let's suppose, one of the twin prime pair is $P_1 ,P_1+2$ and another pair is $P_2, P_2+2$. Where $P_1$ ...
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Smallest twin-prime of the form $k \cdot 11699$#$\pm 1$?

Denote $p$#$:=2\cdot 3\cdot 5\cdot 7\cdots p$ What is the smallest twin-prime of the form $k\cdot 11699$#$\pm 1$ , where $k$ is a positive integer ? Sieving out the candidates with Newpgen and ...
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1answer
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Prime Squares: $(a,b,c,d)$ such that $a,b,c,d,a+b+c+d$ are each between twin primes

[Edited since answer] I have an updated and more advanced version of the prime square in this post: Prime Square: Updated Concept Hello everyone! I would like some feedback on a new idea of mine. It'...
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1answer
109 views

Diophantine equation with application to twin primes

I don't believe one exists, but here's the question: What is the largest $x \in \mathbb{N}$ such that it cannot be represented in any of the following forms $a,b \in \mathbb{N}$... $6ab+a+b-...
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Determining an upper bound for a plot of twin primes

Since this is long and requires a lot of explanation, let me briefly and vaguely state my question at the onset: how would I go about determining an upper bound for the plot pictured near the bottom ...
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1answer
236 views

$pq+1$ is a square $\iff$ $p$ and $q$ are twin primes

This exercise if from Beachy and Blairs Abtract algebra book. Assume that $p$ and $q$ are primes. Show that: $pq+1$ is square $\iff$ $p$ and $q$ are twin primes. The backward direction is: Assume ...
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2answers
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Are all these pairs of primes twin primes?

For $a$ and $b$ primes, if both $(a^b \bmod b)$ and $(b^a \bmod a)$ are prime, does this imply that $(a,b)$ are twin primes? For example, for $(a,b)=(41,43)$, $(41^{43} \bmod 43) = 41$ and $(43^{41} \...
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A simple explanation to this asymmetry?

Let $S_k^\delta=\{a+b\le k\mid(a,b)\in\mathbb N_+^2\wedge a^2+b^2\in\mathbb P\wedge a^2+b^2+\delta\in\mathbb P\}$, where $\mathbb P$ is the set of primes. If $\delta=2$ then the condition on $a^2+b^...
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Odd numbers are of form $a+b$ where $a^2+b^2$ is a twin prime

Conjecture: Any odd natural number $n\notin \{1,27\}$ is of form $n=a+b,\,a,b\in\mathbb N^+$, where $a^2+b^2$ is a twin prime. This is a stronger variant of the conjecture Any odd number is ...
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Translational equilibria between primes and composites.

Think of the primes as bin $P$ and the composites as bin $C$, and let $0 \in P$. We can already say that given $\sigma : \Bbb{Z} \to \Bbb{Z} : x \mapsto x + 2$ $P^C = \{ p \in P : \sigma(p) \in C \} ...
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1answer
157 views

Smallest twin-prime-pair above $2\uparrow\uparrow 5\ $?

I searched the smallest prime larger than $$N:=2\uparrow\uparrow 5=2^{65536}$$ $N$ has $19\ 729$ digits. This is quite large and finding primes of this magnitude is not easy any more. I found $$N+44\ ...
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1answer
91 views

Elementary topological proof of twin prime conjecture.

For any function $f: \Bbb{N}^{\times} \to \Bbb{N}^{\times}$, $|f(n) - f(m)|$ is a pseudometric on $\Bbb{N}^{\times}$. When $f = \Omega$ the number of prime divisors including multiplicity, we get a ...
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0answers
89 views

Can the twin prime conjecture be stated in terms of ideals in a useful way?

Let $\mathcal{I}^{\bullet}(\Bbb{Z})$ be the set of nonzero ideals of $\Bbb{Z}$. It is a cancellative multiplicative monoid since $\Bbb{Z}$ is a PID. Define $\mathcal{I}^{\bullet} \xrightarrow{\phi} \...
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1answer
232 views

Spot the mistakes? Proof of the Twin Prime conjecture and Goldbach's theorem

The Twin Prime Conjecture For any prime number $p_x$ larger than 3, there exists a number $n$ that is less than $p_x^2 -2$ and does not have a remainder of $\pm 1$ when divided by any prime number ...
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135 views

Euler Totient of Numbers Between Twin Primes.

Are there any known special properties of a number located between twin primes? The question came up in the discussion. (The expression below has been rephrased to a weaker form for clarity) In ...