Quote:
Originally Posted by robert44444uk
With the group's current computing resources it should be possible to take the following sequence and extend it
https://oeis.org/A052187
So 47,53,59 is the first sequence of primes with a repeated gap of 6. Repeated gaps are all 0mod6. This is easy to prove.
The highest known members of the sequence are 46186474937633 46186474937957 46186474938281, differing by 324. However no repeating gap of 318 is know below 5e13 according to Johnson and Resta.
I might have a look at 5e13 to 6e13

A minor correction to your original comment: Repeated gaps are all 0mod6
except for the triplet 3, 5, 7.
Also, I believe the name for this is "balanced primes." You can read more about that here.
https://en.wikipedia.org/wiki/Balanced_prime