Articles of prime twins

About a paper by Gold & Tucker (characterizing twin primes)

I’ve carefully looked at the questions on prime and twin prime, but the following question seems not to habe been asked before. Context: In the paper by Jeffrey F. Gold and Don H. Tucker titled A Characterization of Twin Prime Pairs (published in: Proceedings NCUR V. (1991), Vol. I, pp. 362-366, see http://www.math.utah.edu/~gold/doc/char.pdf) the authors […]

Is it cheating to use the sign function when sieving for twin primes?

Let: $$x=2$$ $$t(1,1)=1$$ $$t(2,1)=0$$ $$t(3,1)=0$$ $$\text{ If }n=k \text{ then } t(n,k)= 1$$ $$\text{ if } n>3 \text{ else if } k=1 \text{ then }$$ $$t(n,k)= \text{sgn}((2- \prod _{i=1}^{n-1} t(n,i+k)- \prod _{i=1}^{n-1-x} t(n-x,i+k)) \cdot (2- \prod _{i=1}^{n-1} t(n,i+k)- \prod _{i=1}^{n-1+x} t(n+x,i+k)))$$ $$\text{ else if } n \bmod k=0 \text{ then } t(n,k) = t\left(\frac{n}{k},1\right) \text{ […]

Let $p_k$ be the $k$th prime, can it be shown for $p \ge 5$, that there is not always a twin prime between $p_k^2$ and $p_{k+1}^2$?

For any primorial $p_k \ge 3$, $p_k\#$, there are $$\prod_{2\le{i}\le{k}} (p_i-2)$$ distinct instances of $x,x+2$ that are relatively prime to $p_k\#$. If any of these pairs are less than $p_{k+1}^2$, then they are necessarily twin primes. For the heck of it, I wrote a tiny app that checks all the primes up to 191,137 and […]

Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?

Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?

A conditional asymptotic for $\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$, when $\alpha>-1$

When I’ve followed a notes that show how obtain a similar asymptotic using Abel summation formula, my case with $a_n=\chi(n)$, the characteristic function taking the value 1 if $p$ is prime (in a twin prime-pair, thus caution I’ve defined $\chi(p+2)$ as zero) and $f(x)=x^{\alpha}$, which $\alpha>-1$, and Prime Number Theorem, in my case I am […]

What percentage of numbers is divisible by the set of twin primes?

What percentage of numbers is divisible by the set of twin primes $\{3,5,7,11,13,17,19,29,31\dots\}$ as $N\rightarrow \infty?$ Clarification Taking the first twin prime and creating a set out of its multiples : $\{3,6,9,12,15\dots\}$ and multiplying by $\dfrac{1}{3}$ gives $\mathbb{N}: \{1,2,3,4,5\dots\}.$ This set then represents $\dfrac{1}{3}$ of $\mathbb{N}.$ Taking the first two: $\{3,5\}$ and creating a set […]

First Order Logic: Prove that the infinitely many twin primes conjecture is equivalent to existence of infinite primes

I’m learning First Order Logic independently using a college textbook. I’ve been doing some self exercise question in it and came across this one, which I can’t seem to figure out how to do: Let there be a language $L = \{ +, \cdot, 0, 1, < \}\cup \{ c_{n}\mid n\in \mathbb{N}\}$ and $N$ a […]

Twin Primes (continued research)

This has become increasingly crowded, so at the onset, let me state this: My question is, is there some reason this is so linear that I’m not seeing? The only thing it seems to indicate to me is that there truly must be infinitely many twin primes. I’ve previously posted a method that might have […]

Does proving the following statement equate to proving the twin prime conjecture?

After some research, I found that it has been supposedly proven, that proving that there exists an infinite number of positive integers K such that; $K \neq 6ab \pm a \pm b$ and $K \neq 6ab \mp a \pm b$ implies that the twin prime conjecture is true. My sources include the following; https://oeis.org/A002822, http://arxiv.org/abs/1106.6050 […]

Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang’s paper on prime gaps? Wired reports “$70$ million” at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who can explain the proof? Is the outline in the annals the preprint or the full accepted paper?