I was flipping the math book where I saw a table of primes. The primes were marked in black bold.
It's interesting to see that except 3,13,23, lots of primes (not necessarily consecutive) occur in pairs $p$ and $p+10$, and their distribution compare to other isolated primes didn't seem to reduce in $1000$. So I googled and there was a thing called https://en.wikipedia.org/wiki/Twin_prime .
My question was that was the pair of primes of module 10 just a coincidence?
On
(Too long for a comment.)
Actually, if humans had $12$ fingers and generally used the duodecimal system (base $12$), the results would have been more striking.
I used Mathematica to find the number $N$ of prime pairs $p$ and $p+m$ for the first $10000$ primes and the results are summarized below:
$$\begin{array}{|c|c|c|} \hline m&N&\text{name}\\ \hline 2&1270&\text{twin primes}\\ 4&1264&\text{cousin primes}\\ 6&\color{blue}{2538}&\text{sexy primes}\\ 8&1303&-\\ 10&1682&-\\ 12&\color{blue}{2515}&-\\ 14&1546&-\\ \hline \end{array}$$
The fact that all primes $p>3$ have form $6n\pm1$ may explain the preference for $m=6$ and $m=12.$
P.S. The interesting name "sexy prime" has to do with the Latin word for six (sex), though whoever coined it may have had other things in mind.
Here is my version of the table that Tito Piezas III presented in his answer. Initially I wrote the wrong code (using computer algebra Reduce) and wondered why I didn't get the same numbers as in Tito's table, then I realized that I was computing the number of pairs, for the first 10000 primes $p\ge3$ such that the next prime was a given distance $m$ from $p$ (e.g., distance $2,4, ...,36$). This number of primes is in column $K$ in the table below (whereas column $N$ is the number of pairs, for the first 10000 primes $p\ge3$ such that $p+m$ is also a prime). For instance, the prime $101$ is counted, when $m=6$, in column $N$ since $101+6=107$ is also a prime, but not counted in column $K$ since the next prime after $101$ is $103$ which is not distance $6$ from $101.$
\begin{array}{|c|c|c|} \hline m&N&K\\ \hline 2&1270&1270\\ 4&1264&1263\\ 6&\color{blue}{2538}&\color{blue}{2012}\\ 8&1303&801\\ 10&1682&953\\ 12&\color{blue}{2515}&\color{green}{1008}\\ 14&1546&513\\ 16&1275&354\\ 18&\color{blue}{2569}&\color{green}{537}\\ 20&1701&249\\ 22&1403&235\\ 24&\color{blue}{2578}&\color{green}{222}\\ 26&1402&91\\ 28&1519&102\\ 30&\color{red}{3451}&\color{green}{154}\\ 32&1246&35\\ 34&1357&36\\ 36&\color{blue}{2561}&\color{green}{55}\\ \hline \end{array}
It seems interesting that $N$ is about the same for all $m$ that are multiples of $6$, up to $m=36$, except for a spike at $m=30$ (this may have something to do with $30$ being divisible by the small prime $5$ (in addition to $2$ and $3$) whereas $6,12,18,24,36$ each is divisible only by primes $2,3$). I feel that a form of the pigeon-hole principle is relevant, if you put this many primes into this little space, some distances between these primes ought to repeat.