Sexy prime quintuplets

If $p = 5$ then each of $p + 6 = 11, p + 12 = 17, p + 18 = 23$ and $p + 24 = 29$ is prime. To show that this is the unique solution, consider any prime $p$ that is not divisible by $5$ . Then $p$ is equivalent to one of $1,2,3,4$ modulo $5$ .

• if $p \equiv 1 \pmod{5}$ then $p + 24 \equiv 0 \pmod{5}$ and is thus not prime

• if $p \equiv 2 \pmod{5}$ then $p + 18 \equiv 0 \pmod{5}$ and is thus not prime

• if $p \equiv 3 \pmod{5}$ then $p + 12 \equiv 0 \pmod{5}$ and is thus not prime

• if $p \equiv 4 \pmod{5}$ then $p + 6 \equiv 0 \pmod{5}$ and is thus not prime

So if $p$ is any prime other than $5$ then one of $p + 6, p + 12, p + 18$ or $p + 24$ will be divisible by $5$ , proving that $p = \boxed{5}$ is the unique solution.

We early see that $6,12,18,24$ are all congruent to different values mod $5$ , and neither of them is congruent to $0$ . Thus one of the $4$ sums will be divisible by $5$ if $p\not \equiv 0 \mod 5$ . But $p$ is prime, thus $p=5$