The prime gap list project is a community curation and continuation of the contribution of Dr. Thomas Ray Nicely who, for nearly 20 years, maintained a reference web site which included a list of all first known occurrence prime gaps (see Prime Gap on wikipedia). Dr. Nicely acted as a central point of contact for the submission and publishing of new record prime gaps.
Dr. Nicely died in a car accident in Sept. 2019 and the Prime Gap Searches group of the Mersenne forum has taken on the responsibility of curating the list of all first known occurrence gaps and processing submissions of new record prime gaps via a community-maintained Github repository.
The purpose of the forum is to act as a discussion area and an information exchange for those people interested in prime gaps.
We have a number of active searches going on, but all of the searches have something in common, the search for candidates that qualify to be listed on Dr Thomas Nicely’s website http://www.trnicely.net/gaps/gaplist.html
The Nicely site keeps records of gaps of a specified length between two integers that are proven prime or shown to be probable primes. Each integer between the two end points is proven to be composite.
This forum is particularly interested in prime gaps that have a merit of great than 10. For a given prime , the average prime gap to the next prime is given by , where is the natural logarithm. The merit of a gap is given by .
There is little point in looking for gaps <1,352 as an exhaustive search of primes up to has been carried out and all gaps smaller than this have been found.
As of the summer of 2014, the Nicely site had early instance prime gaps with merit >10 listed for all possible gaps <60,000 and an early effort by this forum has been to extend the early instance list up to 100,000.
At the far end of the scale, the forum is helping to support the largest gap search, looking at a candidate gap (4,680,156) provide by mart_r. This has a merit >20.
The forum also is looking at new approaches for finding large merit gaps. For example, a theoretical approach that looks at the modular relations of small primes over a range of 2310 integers that might provide record gaps in the range 1,350-4,000.
Last, but not least, the forum looks to develop gap hunting programs that can help to find those large merit gaps. - robert44444uk (Rob Smith)