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I am interested in the minimal gap between prime sextuplets. A prime sextuplet is a cluster of $6$ primes of the form $p, p+4, p+6, p+10, p+12, p+16$. (See OEIS A022008.) For example, $97, 101, 103, 107, 109,$ and $113$ form a prime sextuplet.

With exceptions at the very beginning of the sequence of primes, an interval of $17$ consecutive numbers can contain at most $6$ primes, due to considerations mods $2, 3, 5$. And so, a prime sextuplet gives you a maximum number of primes within an interval of $17$ consecutive numbers.

If you also consider mod $7$, you find that in avoiding multiples of $7$, the primes in a sextuplet must be of the form $210k + 97, 210k + 101, 210k + 103, 210k + 107, 210k + 109$, and $210k + 113$. There is also one “exceptional” prime sextuplet consisting of $7, 11, 13, 17, 19, 23$. It is "exceptional" because it includes a multiple of $7$, namely $7$ itself. But of course, all subsequent prime sextuplets must avoid multiples of $7$, and then the above form holds.

I am interested in how close together (nonexceptional) prime sextuplets can be. In a computer search up to $8\cdot 10^9$, the closest pair of sextuplets that I found had leading elements that differed by $2730$ (the leading primes in these two sextuplets were $2\_859\_875\_647$ and $2\_859\_878\_377$). But in principle it seems that (at least using mod considerations) there could be prime sextuplets whose leading primes differ by as little as $210$.

So, my question is: Are there two nonexceptional prime sextuplets whose leading primes differ by $210$? I am also interested in two nonexceptional prime sextuplets whose leading primes differ by less than $2730$.


Remarks: (1) As a comparison, a prime quadruplet is a set of four primes of the form $p, p+2, p+6$, and $p+8$. Prime quadruplets have the form $30k + 11, 30k + 13, 30k + 17, 30k + 19$. One might hope to find two prime quadruplets whose leading primes differ by $30$. And indeed, with patience, you can find such prime quadruplets, the first instance having leading primes of $1\_006\_301$ and $1\_006\_331$ (see OEIS A059925.) This first double quadruplet is perhaps further out than might be expected, given that double quadruplets exist at all. This gives me hope that there are prime sextuplets somewhere out there that are closer together than $2730$.

(2) Background on the question: I have, since quite early in my mathematical journey, been interested in dense clusters of prime numbers, and have come back to these questions periodically over the last few decades.

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  • $\begingroup$ In principle it should be possible. How small will it be? That's another question entirely. Finding the answer likely depends on your available computing power. I wouldn't be surprised if the first such sextuples are into the $10^{12}$ range or higher. $\endgroup$ Jan 20 at 23:01

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The prime $k$-tuples conjecture asserts that there are infinitely many integers $n$ such that $n+a$ is prime for all $a\in\{0, 4, 6, 10, 12, 16, 210, 214, 216, 220, 222, 226\}$, since this set of $12$ integers omits at least one residue class modulo $p$ for each prime $p\le12$. Note that such a configuration of primes is the union of two prime sextuples $210$ apart, which would imply that the answer to the OP's question is yes.

(One can check that every smaller offset results in a set that covers all residue classes modulo one of $2$, $3$, $5$, or $7$, so this is the most compact possible union of two sextuples.)

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  • $\begingroup$ Yes, but the prime $k$-tuples conjecture is just a conjecture. Also, I would be quite interested in specific numerical examples. $\endgroup$
    – paw88789
    Jan 20 at 18:58

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