One of the challenges of the prime gap pages is making sure the submitted gaps are correct. Ideally there would be a check on the bounding integers: are they prime? And the intermittent numbers: are they composite? Ideally only after a double check confirms the submitted gap, would it be accepted.
In the real world the larger the starting value of the gap the longer and longer the certification of the bounding integers takes. The current record gap with proven bounding primes is a gap of 1113106, proving the bounding integers (both with 18,662 digits), are prime took over 6 weeks with 12 threads. For gaps of this size and larger it’s infeasible for the server to certify the bounding integers let alone the compositeness of the intermittent/interal integers.
The classification of a gap is meant to express the level of certainty has that the gap is valid; meaning both bounds are prime and there are no intermediate primes. A mix of certification, verification, spot checking, double check, and trust of submitters forms the basis for this lists and the classification.
There are different levels of certainty the gap entry code can award the gaps that are submitted. The combinations can be summarized in the following table:
|Bounding integers ->||Certified||Double Checked||Trusted discoverer|
|If internal gap double-checked||C*||D*||T|
* Only these classifications are regurarly used
Table 1: The levels of certainty of prime gaps
The highest level of certainty is achieved when the bounding integers are certified (e.g. using ECPP, FastECPP, ECPP-DJ, or Primo and ideally submitted to factordb) and the interal integers are double checked as composite (e.g. using OpenPfgw, Pari/GP, etc.). If this is the case, a capital “C” will be used as the classification. In the uncommon situation that the bounding integers are certified, but no double check has been performed on the internal gap, a lower-case “c” will be used as the classification.
The majority of the gaps will have PRP performed on the endpoints and either a complete or partial double check on the internal gap. These will be classified “D” or “d” depending on if the internal gap was fully double checked.
Very large gaps (think top 20 sized gaps) can take too long to check these will be added with the lowest classification, lower-case “d” (the server always checks the endpoints) or manually as “t”. If the internal gap is fully checked later, the classification can be upgraded to “D”.
For gaps less than <= 100,000 * the server performs a PRP (probable prime) test using libGMP or PFGW on the endpoints and on all internal integers (after an aggressive sieving process). These gaps are marked as having been double checked with a capital “D”.
For medium sized gaps (100,000 < gap <= 300,000) only some of the internal integers are spot-checked and the range is marked as “d”.
For gaps > 300,000 the code only checks the bounding integers but requires the discover be trusted. These gaps are marked “d” and can be upgraded later (see next section).
At this time the server lacks ECPP capability and can’t certify endpoints. In the future small gaps (think gap <= 5000) will have bounding primes certified prime and all intermittent integers checked. This is the highest level of certainty that can be achieved and these gaps will be marked with a capital “C”.
For small gaps Seth can manually run Sage’s ECPP using
For large gaps (with high merit or that have reached mega-gap status) it’s easiest to ask for formal double check on the MersenneForum thread .