# 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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 $N$ ...