A conjecture on the closeness of twin primes Let $p_1$ and $p_2$ be twin primes, and let $p_1-1=a_1\times b_1$ and $p_2+1=a_2\times b_2$ be such that $|b_1-a_1|$ and $|b_2-a_2|$ are minimised. Similarly,  let $p_1+1=p_2-1=a\times b$ be such that $|b-a|$ is minimised.
Now assume the twin prime conjecture, and let $$\mathcal P_n=\frac{\#\text{twin primes} \le n:\min\{|b_1-a_1|,|b-a|,|b_2-a_2|\}\neq|b-a|}{\#\text{twin primes} \le n}$$ What is the value of $\mathcal P_\infty$?
Note that by convention the pair $(2,3)$ are not twin primes.

For twin primes less than $100$, we have that $p_1=5,17$. Hence I think that if there are infinitely many such primes then they will be exceedingly sparse.
Here is a table showing $\mathcal P_n$ for increasing values of $n$. Credits to Enzo Creti for running the program. $$\small\begin{array}{c|c}\log_{10}n&amp;1&amp;2&amp;3&amp;4&amp;5&amp;6&amp;7&amp;8&amp;9&amp;10&amp;11\\\hline \mathcal P_n&amp;\frac12&amp;\frac14&amp;\frac{17}{35}&amp;\frac{93}{205}&amp;\frac{600}{1224}&amp;\frac{4326}{8169}&amp;\frac{31939}{58980}&amp;\frac{243876}{440312}&amp;\frac{1928700}{3424506}&amp;\frac{15661079}{27412679}&amp;\frac{129632703}{224376048}\\\hline\text{decimal}&amp;\small 0.5&amp;\small0.25&amp;\small0.4857&amp;\small0.4537&amp;\small0.4902&amp;\small0.5296&amp;\small0.5415&amp;\small0.5539&amp;\small0.5632&amp;\small0.5713&amp;\small0.5777\end{array}$$
This can be seen more clearly in the following plot; more interesting is the behaviour after $n=10^4$.

There is a slight dip at $n=10^2,10^4$ but otherwise it looks like there is a converging increase in $\mathcal P_n$. It is unlikely, but if there reaches a point where $\mathcal P_k=\mathcal P_{k+1}$ then the twin prime conjecture will be disproved.
Twin primes other than $3,5$ are of the form $6k\pm1$, which makes $p_1-1 = 6k-2$, $p_2 + 1 = 6k+2$, and $p_1+1 = p_2-1 = 6k$. Define $f(x) = \min\{|a-b|\}$ where $a b = x$, and you are asking are there infinitely many $k$ where $6k\pm1$ are both primes, and $f(6k-2) &lt; f(6k)$ or $f(6k+2) &lt; f(6k)$. 

Experimentally, about $54\%$ of the twin primes are "close", not the $33\frac13\%$ that one would naively expect. Still, most likely infinitely many.
Twin primes other than $3,5$ are of the form $6k\pm1$, which makes $p_1-1 = 6k-2$, $p_2 + 1 = 6k+2$, and $p_1+1 = p_2-1 = 6k$. Define $f(x) = \min\{|a-b|\}$ where $a b = x$, and you are asking are there infinitely many $k$ where $6k\pm1$ are both primes, and $f(6k-2) &lt; f(6k)$ or $f(6k+2) &lt; f(6k)$. 

Experimentally, about $54\%$ of the twin primes are "close", not the $33\frac13\%$ that one would naively expect. Still, most likely infinitely many.