# Questions tagged [twin-primes]

For questions on prime twins.

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### Prove that at least two are the same $p = b^c + a , q = a^b + c , r = c^a + b$

Given a, b, c ∈ N p = bc + a , q = ab + c , r = ca + b we know that p q r are primes. Prove that at least two of the p ,q ,r are the same. Edit: i have tried with contradiction method.I assumed all ...
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### An argument for the infinitude of twin primes

Beyond 3, potential prime numbers occur at values equal to $6n\pm1$ where $n=1,2,3...\infty$. When both of these values are prime for a given $n$, a twin prime pair occurs. Dividing the number line ...
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### Prime twin counting by $\pi_2(t^2) =^? \sum_{2<j<t^2} (-2)^{\omega(j)} (1/2)(\lfloor{\frac{t^2}{j}}\rfloor +\lfloor{\frac{t^2-2}{j}}\rfloor) +C$?

Let $\omega(n)$ count the number of distinct prime factors of the integer $n \geq 2$. This $\omega(n)$ is called the prime omega function. Inspired by these ideas : Improved sieve for primes and ...
1 vote
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### question on an extension of a theorem of Mertens

Mertens' theorem 3 states: $$\lim_{n\to\infty}\prod_{p\le n}\left(1-\frac{1}{p}\right)=\frac{e^{-\gamma}}{\log n}$$ Separately, the twin prime constant $C_2$ is defined as: \begin{align} C_2 &= \...
1 vote
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### About prime gaps ; $\sum_{m = 2}^{\ln_2^{*}(n)} \pi_{m}(n) \leq \pi(n)$?

Let $\pi(n)$ be the number of primes between $1$ and $n$. Let $\pi_2(n)$ be the number of prime twins (gap $2$) between $1$ and $n$. Let $\pi_3(n)$ be the number of prime cousins (gap $4$) between $1$ ...
1 vote
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### Is there any twin prime representing function?

There are many prime representing functions. For example, $\lfloor A^{3^n} \rfloor$ is prime representing function because for all positive integers n ,it generates a different prime number. Here $A$ ...
1 vote
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### If $n$ is a Poulet number then $n+2$ is not a Poulet number?

Consider Fermat pseudoprimes to base $2$, also called Sarrus numbers or Poulet numbers. Inspired by prime twins it makes sense to consider : Conjecture : If $n$ is a Poulet number then $n+2$ is not a ...
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### An interesting finding on twin primes

I was doing some research on prime gaps including twin primes and this led me to this finding, which is: $$\lim\limits_{n\to \infty} \frac{\pi^2(n)}{n\pi_2(n)} = 0.7550363087870907 \cdots\cdots (1)$$...
1 vote
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### All twin prime averages in the range $[9, 119]$ are of the form $6(5[3(z-x)]_{\pmod 7} + x)$ for some $x \in \{0,2,3\}, z \in \{0,2,3,4,5\}$.

Question. Can we come up with a general formula $f(x_5, x_7, x_{11}, \dots, x_{p_n})$ such that each twin prime average $a \in [p_n + 2, p_{n+1}^2 - 2]$ is expressible as $f(x_5, \dots, x_{p_n})$ for ...
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### $n+1$ and $n \phi (n) + 1$ are both perfect squares if and only if $n$ is a product of twin primes?

I'm trying to prove the following conjecture concerning twin primes and Euler's totient function, which I have verified for $n$ up to 1 billion. For all $n \in \mathbb{N}$, $n+1$ and $n \phi (n) + 1$ ...
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### $A(x) = \prod_{p,q \in \Bbb{P}} (1 - \frac{x^2 - 1}{pq}) = 0$ if and only if $x$ is a twin prime average?

Conjecture. For any integer $x \geq 1$, we have: $$A(x) = \prod_{p,q \in \Bbb{P} \\ p \lt q} \left(1 - \frac{x^2 - 1}{pq}\right) = 0$$ if and only if $x$ is a twin prime average. How can we prove ...
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### A corollary of Brun's pure sieve from Opera de Cribro by Friedlander and Iwaniec

I'm stuck on the one (if not many) step of the proof of Corollary 6.2 (page 58) from the book "Opera de Cribro" by Friedlander and Iwaniec. The statement is a corrollary of Brun's pure sieve ...
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### Potential infinite, fast growing subsequence of twin primes

Probably the most interesting part of this discussion is about twin primes of the form $6x\pm 1$, with $x=4\cdot(5\cdot 7\cdot 11\cdot 13\cdot 17 \cdot 19\cdot 23)$ being a typical example, and the ...
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### Observation about twin primes: is it true? If so, why?

I noticed today that every set of twin primes except for $(3,5)$ and $(5,7)$ seems to have one of the two primes that can be represented by the sum of two squares. For example: \begin{eqnarray*} 13=3^...
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### For twin primes $p$ and $q$, prove there is an integer $a$ such that $p|(a^2-q)$ if and only if there is an integer $b$ such that $q|(b^2-p)$.
For twin primes $p$ and $q$, prove there is an integer $a$ such that $p|(a^2-q)$ if and only if there is an integer $b$ such that $q|(b^2-p)$. Algebraic substitution using $p=q+2$ and the definition ...