Questions tagged [twin-primes]
For questions on prime twins.
314 questions
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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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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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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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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 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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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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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 ...
user243301
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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 $P_1$ ...
user508245
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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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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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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 $a,b \in \mathbb{N}$...
$6ab+a+b-1$...
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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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$pq+1$ 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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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 of ...
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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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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 $$N+44\ ...
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Elementary topological proof of twin prime conjecture.
For any function $f: \Bbb{N}^{\times} \to \Bbb{N}^{\times}$, $|f(n) - f(m)|$ is a pseudometric on $\Bbb{N}^{\times}$. When $f = \Omega$ the number of prime divisors including multiplicity, we get a ...
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Can the twin prime conjecture be stated in terms of ideals in a useful way?
Let $\mathcal{I}^{\bullet}(\Bbb{Z})$ be the set of nonzero ideals of $\Bbb{Z}$. It is a cancellative multiplicative monoid since $\Bbb{Z}$ is a PID. Define $\mathcal{I}^{\bullet} \xrightarrow{\phi} \...
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Spot the mistakes? Proof of the Twin Prime conjecture and Goldbach's theorem
The Twin Prime Conjecture
For any prime number $p_x$ larger than 3, there exists a number $n$ that is less than $p_x^2 -2$ and does not have a remainder of $\pm 1$ when divided by any prime number ...
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Euler Totient of Numbers Between Twin Primes.
Are there any known special properties of a number located between twin primes?
The question came up in the discussion. (The expression below has been rephrased to a weaker form for clarity)
In ...
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What is the longest sequence of consecutive twin primes known to exist?
For example a Prime triplet where each prime number is separated by a single even number is:
$$3, 5, 7$$
it contains 3 prime numbers each separated by 2
What is the longest such sequence known to ...
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Is this an artifact of graphing prime number ratios, or a relationship between twin primes?
I started with this question about linking at least one twin prime with each prime. My approach is:
Give each prime two buckets.
Put each odd 3n in the second bucket of it's largest prime divisor.
Put ...
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Is this an unknown pattern in prime numbers?
I am trying to figure out if the pattern I've found concerning twin primes is a known pattern or not. It turns out that with every set of twin primes, if the higher of the two numbers is converted to ...
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New lower limit for the number of twin primes $\pi_2(n)$
While trying to find a lower limit for the number of twin primes I noticed the problem of having to compensate for duplicates. Once I overcame this problem the duplicates of the duplicates became a ...
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Is this a new twin prime sieve?
I developed a twin prime sieve and would like to know if there is a well known equivalent. Pardon my lack of proper notation. Feedback on how to format this properly would also be appreciated.
...
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Asymptotic expressions of $\pi_{2}(n), \pi_{4}(n)$ and $\pi_{6}(n)$
Let $n$ be a natural number. Let $\pi_k$ be denoted as follows.
$ \pi_{2}(n) = $ the number of twin primes $(p, p+2)$ with $p \le n$.
$ \pi_{4}(n) = $ the number of cousins primes $(p, p+4)$ with $p ...
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Proof of minor claim related to the Twin Primes Conjecture
Question:
How can it be proven that integers of the form $n=6jk\pm j \pm k;\ j,k\in \mathbb N^*,$ are the only ones which (when multiplied by $6$) correspond to multiples of $6$ not between twin ...
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On the asymptotic behaviour as $X\to\infty$ of the sum $\sum\frac{1}{x}$ over every $x\leq X$ such that $\pi_2(x)<2C_2\int_2^x\frac{dt}{\log^2t} $
Let $\pi_2(x)$ the twin prime-counting function that counts the number of twin primes $p,p+2$ with $p\leq x$, and $C_2$ is the the twin prime constant. We assume the Twin Prime conjecture, see it as ...
user243301
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Is it possible a Bonse's inequality for twin primes, on assumption of a form of the Twin Prime conjecture?
This Wikipedia article shows us what are the first twin primes. Here we state the notation for each twin prime pair writing $(q_n,2+q_n)$ for $n\geq 1$ (for example $q_1=3$ and $q_8=71$).
On the ...
user243301
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about infinitely many bounded prime gaps
Let $p_n$ be the $n$'th prime $\alpha \in (0,1)$ and $e$ the euler constant. By mean value theorem one has: $$ e^{p_{n+1}^{\alpha}} = e^{c_n^{\alpha}}\frac{\alpha}{c_n^{1 - \alpha}} \cdot (p_{n+1} - ...
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Twin Primes by an amateur mathematician
About 20 years ago in a bookstore of Tokyo I found a book titled "Twin Primes" by Seiji Yasui, an amateur mathematician. He insisted that he proved the infinity of twin primes.
The proof is attached ...
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Prime Number Congruence Conjecture
$6±1 = (5,7)$
5 and 7 are twin primes
$6 + 5 = 11$
$6 + 7 = 13$
11 and 13 are twin primes
$ 6 × 5 = 30 $
$ 6 × 7 = 42 $
30 and 42 are adjacent to the twin primes (29. 31) and (41, 43)
Is there ...
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How many twin primes are of the form $2^n-1$ and $2^n+1$?
The first pair is $(3,5)$ for $n=2$. Is there any other pair beside this?
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Longest prime $k$-tuplet such that each prime $p$, $\gcd(p-1, 420) > 2$?
What is the longest possible length $k$ of a prime $k$-tuplet (smallest groupings of $k$ primes) such that each prime $p$ in the $k$-tuplet,$\gcd(p-1, 420) > 2$. In other words, each prime $p$ in ...
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$n$ such $n!+1$ and $n!-1$ are both primes
I am looking for positive integers $n$ such that $n!+1$ and $n!-1$ are both primes.
Looking in OEIS Lists, I found that the only known $n$ such $n!+1$ and $n!-1$ are both primes is $n=3$ (giving $5$ ...
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Twin prime conjecture and finite groups. Help with proof of basic equivalence. [closed]
Conjecture: When $n \gt 2$ is an even number, $n\pm 1$ is a twin prime pair iff there exists a nontrivial solution of $X^{(n-2)n} = 1 \pmod {n^2 - 1}$.
Proof:
To see this suppose $n \pm 1 \in \Bbb{...
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Is it cheating to use the sign function when sieving for twin primes?
Let:
$$x=2$$
$$t(1,1)=1$$
$$t(2,1)=0$$
$$t(3,1)=0$$
$$\text{ If }n=k \text{ then } t(n,k)= 1$$
$$\text{ if } n>3 \text{ else if } k=1 \text{ then }$$
$$t(n,k)= \text{sgn}((2- \prod _{i=1}^{n-1}...
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Proving two inequalities that are used in the proof of Brun's theorem
So I was reading a proof of Brun's theorem in AoPS, then I came to these two inequalities that I can not figure out how they have been obtained.
It states that if $p$ stands for prime numbers in the ...
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Conjecture on concatenation of twin prime numbers
This question is bourn out of my recent answer to a question asked here
can a Car Registration Number, a combination of prime, be prime?
Now this is true for certain prime numbers , e.g., $3, 7, 109$...
user356774
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Does Zhang's work imply that the next greatest prime number is necessarily less than $2^{57,885,161}-1+ 70000000$?
Actually the greatest known prime is $2^{57,885,161}-1$. Zhang's work proved that the interval between pairs of prime numbers is limited to $70,000,000$. Does it imply that the next greatest prime ...
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Infiniteness of twin prime powers
This question is a weaker version of the twin prime problem.
Question: Are there infinitely many prime powers $p^a, q^b$ with $p^a-q^b=2$?
Of course, we expect an answer easier than solving the ...
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