It is suspected that there are infinitely many prime quintuplets, although this has never been proven.

The largest known prime quintuplet at the time when it was found on December 2020 is equal to (p, p + 4, p + 6, p + 10, p + 12), where p = 566761969187  ·  4733# / 2 − 8 (where # denotes the primorial).[1]

The largest known prime quintuplet of the form (p, p + 2, p + 6, p + 8, p + 12) at the time when it was found on November 2020 is equal to p = 126831252923413 · 4657#/273 + 1.[1]

Sources

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