Questions tagged [prime-twins]

For questions on prime twins.

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Why is the twin prime conjecture hard?

If $\pi_2(x)$ is the number of twin primes of magnitude less than or equal to $x$. We want to prove that $$\lim_{x\,\to\,\infty}\pi_2(x)=\infty$$ which should be easier than finding and proving an ...
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38 views

The distribution of numbers coprime to $p_k\#$ and the implications for the occurrence of twin primes

Consider the interval $[p_k,p_k\#]$. It contains $(\phi(p_k\#))-1=(\prod_{i=1}^k (p_i -1))-1$ numbers relatively prime to $p_k\#$ (NB the $-1$ at the end of each expression is strictly necessary ...
6
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104 views

Goldbach twin and cousin primes

Good day. My question is a counterexample for the following: Is every even number greater than 4 the sum of a number that belongs to set the cousin primes with another number that belongs to the set ...
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2answers
71 views

Infinite primes history

I am little confused with who was the first to modify Euclid's argument of infinitude of primes from $p_{1}p_{2}...p_{r}+1$ to $p_{1}p_{2}...p_{r}-1$? Some writers say it was E.E. Kummer ,($1878$) (...
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A totient-like function relating to twin pairs

In this question Larry Freeman showed that $\prod_1^n(p_i-2)$ reports the number of pairs of numbers $x,(x+2)$ such that $x<p_n\# \wedge \gcd (x(x+2),p_n\#)=1$. This bears a striking resemblance ...
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32 views

Question related to the expression of prime, twin-prime, and Sophie Germain prime counting functions in terms of Mertens function

This question assumes the following definitions. (1) $\quad\pi(x)==\sum\limits_{p\le x}1\qquad\text{(prime counting function where $p\in P$ is a prime})$ (2) $\quad\pi_2(x)==\sum\limits_{p_2\le x}1\...
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Defined intervals conjectured to contain at least one pair of twin primes

Apologies up front for a lengthy post. The questions posed are simple, but the thoughts underpinning them require careful exposition. Definitions: The twin prime pair $(3,5)$, not being of the form $...
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38 views

how to predict the density of these prime numbers?

if I wanna know how many prime number $p$ less than $x$, such that : there is at least one prime number of this form$ 2Kp+1$ where $K={1,2,3,4,5,6,7,8,9,10}$. for example: if $x=1000$, then the ...
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129 views

On a conjecture about the arithmetic function that counts the number of twin primes

I've asked the same question on MathOverflow two days ago as On a conjecture about the arithmetic function that counts the number of twin primes, I add this reference while I hope to know what about ...
2
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1answer
132 views

What is the next nice number?

For $p_1, p_2, p_3, p_4$ be four consecutive prime numbers, we define $$S(p_1, p_2, p_3, p_4) = p_1^2 + p_2^2 + p_3^2 + p_4^2 .$$ We have now: $$2020 = S(17,19,23,29)$$ $$2692 = S(19,23, 29, 31)$$ $$...
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1answer
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What about twin primes in other residue classes?

So as I understand it, the current state of the twin primes problem is that an unconditional proof of infinitely many pairs of primes separated by 246 exists, and I think a conditional proof of pairs ...
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46 views

Is there any known result on twin primes in arithmetic progression?

Is there any general conjecture / asymptote for twin primes in arithmetic progression in analytic number theory apart from Mikawa's paper? This is the paper link https://projecteuclid.org/euclid.tkbjm/...
6
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1answer
142 views

Is the series $X =\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+..$ convergent or divergent.

The series is the reciprocal of twin primes. Let $Y=(y_n)$be the series of reciprocal of natural numbers. Now if I use the comparison test we can see that each term of $0 <(x_n) < (y_n)$ .So ...
8
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4answers
270 views

What is notable about the composite numbers between twin primes?

Look at the composites between twin primes (A014574): $$ 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, \\ 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, \ldots \;...
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2answers
70 views

There is no natural polynomial map (other than $1$) that can eventually leave the $k$-semiprimes behind.

Let $F(S)$ be the free commutative monoid on countably many symbols $S$. Then it's obvious that $F(S) = \{1\} \uplus S \uplus S^2 \uplus \dots$ One can take $S = $ the prime numbers in $\Bbb{N}$ in ...
6
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1answer
104 views

Heuristics of counting twin primes

I was reading on counting the number of twin primes and I found this heuristic explanation on the Hardy-Littlewood conjecture, which states that $$\pi_2(x)\sim 2\Pi_2 \frac{x}{\log^2(x)},$$ where $\...
2
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1answer
65 views

Congruences with twin prime numbers

Let $p$ and $q$ be a pair of twin primes, such that $q = p + 2$. Prove the following: $\exists$ an integer $a$ such that $p \mid (a^2 - q)$ $\iff$ $\exists$ an integer $b$ such that $q \mid (b^2-...
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2answers
626 views

Whats wrong with this model theoretic proof of the twin primes conjecture?

I have a proof of the twin primes conjecture using the compactness theorem. It cannot be correct, because it is too simple. Please help find the flaw. Proof by contradiction, Assumption: there are ...
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24 views

Proven Upper bounds for the number of prime pairs of gap t less than n

I had asked this previously but it was closed as a duplicate. The problem was that people just linked to conjectures. For twin primes they have the upper limit for number of twin primes less than $N$ ...
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Upper bounds on number of cousin primes less than n [duplicate]

For twin primes they have the upper limit for number of twin primes less than $N$ as $C \cdot \frac{N}{{log(N)}^2}$ . Is there similar bounds for cousin primes and other prime pairs with a gap of $...
1
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1answer
32 views

Brun's paper on convergent sum of reciprocals of twin primes

Does anyone know where I can find a PDF (or equivalent) of Brun's 1919 paper (linked to in the references here, but with a title too messy to write out in full on this forum) that proved that the sum ...
3
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42 views

Elementary proofs for weaker statements than the Twin Prime Conjecture

I know that the closest proven statement to the Twin Prime Conjecture is Chen's theorem, stating that there is an infinite number of primes $p$ such that $p+2$ is either prime or semiprime. The proof ...
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1answer
34 views

Is $\pi(N+2)\sim \sum_{p_{n+2}\in\Bbb P}^{N+2} p_n^{\frac{1}{\log(p_{n+2})}}?$

Is$$\pi(N+2)\sim \sum_{p_{n+2}\in\Bbb P}^{N+2} p_n^{\frac{1}{\log(p_{n+2})}}=3^{\frac{1}{\log(5)}}+5^{\frac{1}{\log(7)}}+11^{\frac{1}{\log(13)}}+17^{\frac{1}{\log(19)}}+...+N^{\frac{1}{\log(N+2)}}$$ ...
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50 views

What is known about this twin prime-power counting function?

This question is motivated by the MathOverflow question List of properties of Twin primes Dirichlet series and assumes the following definition of the twin prime-power counting function $k_2(x)$ ...
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0answers
66 views

Prime twins $ (3^n - 2, 3^n - 4) $ conjecture

Let $n$ be a positive integer. Conjecture There are infinitely many prime twins of the form $$ ( 3^n - 2, 3^n - 4) $$ Examples include $$(3^2 - 2,3^2 - 4) = ( 7,5 ) $$ $$ ( 3^{37} - 2 , 3^{37} - ...
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1answer
79 views

My attempt to prove the twin prime conjecture

I want to share with my attempt to prove the infinity of twin primes, and i want to have you opinion: Every odd composite $N \geq9$ can be written as: $p^2_{n}+2p_{n}c=N$ With $c \in Z^+$ and $p \...
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2answers
148 views

Have I just proven the Twin Prime Conjecture? [closed]

For those who aren't familiar with it, the Twin Prime Conjecture wonders if there are an infinite number of prime coordinate pairs $(p, q)$ such that $p=q+2$. I'm wondering if I've proven the Twin ...
13
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1answer
411 views

Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes?

Thinking about Goldbach conjecture, I have the following question: Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes? For ...
3
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0answers
28 views

Another question about numbers of the form $p_i\#+p_{n+1}$

This is a follow on question to the question posted by Peter (Can we always find a prime of the form $\ p$#$+q\ $?). Replacing his variable $q$ with $p_{n+1}$, his conjecture in that question can be ...
4
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3answers
272 views

Can anyone come up with an interesting consequence of the Twin Prime Conjecture being true?

The question is in the title. Was wondering if there are statements equivalent to or a consequence of the statement that there are infinitely many twin primes. If not, then why is this conjecture a "...
5
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1answer
68 views

(m,m+2) is twin prime, iff 4((m−1)!+1)≡−m(mod m(m+2))

I'm a programmer, a newbie on math. I'm trying to code to list twin prime. I've found this: $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$ The pair (m, m + 2) is twin prime,...
0
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1answer
136 views

Probability that $2n$ has no Goldbach partitions.

I'm trying to evaluate the probability that some even integer $2n$ has no Goldbach partitions using the following approach... First, visualize the distribution of primes from $1$ to $n$ as a binary ...
2
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1answer
87 views

What do Wikipedia's statements about the twin prime conjecture really mean?

According to Wikipedia the twin prime conjecture is a special case of the first Hardy-Littlewood conjecture: $$\pi_2(n)\sim 2C_2\frac{x}{(\ln x)^2} \sim 2C_2\int_2^n\frac{dt}{(\ln t)^2}$$ where $C_2$ ...
4
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1answer
110 views

Closed form for totient related product

Euler's totient function can be formulated involving a product of the form $\prod\left(1-\frac{1}{p}\right)$. In particular, if every prime is included in the product, the product can be stated in a ...
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2answers
83 views

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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2answers
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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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1answer
63 views

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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318 views

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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2answers
66 views

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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3answers
59 views

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 ...
2
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3answers
119 views

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 $x-y=p_{n+1}$ ...
3
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0answers
118 views

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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2answers
117 views

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 ...
5
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1answer
152 views

Twin Prime Formula

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 ...
4
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1answer
110 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/...
2
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1answer
85 views

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 ...
3
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1answer
94 views

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}...
2
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0answers
65 views

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
87 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 $m=\prod_{n=1}^{N}Pn<P_{N+1}^{2}$, ...
2
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1answer
49 views

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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