I have found what may be the largest explicitly known gap between
two consecutive primes. The bounding (probable) primes are 2^8191 +
19506585 and 2^8191 + 19559063 and the corresponding gap (defined as
the difference between the primes) is g = 52478. The largest
previously known gap was 50206, according to several sources.
I sieved the first 20 million numbers of 2^13 bits using the first
10^7 primes. This sieved out all the composites with a prime factor
less than or equal to 179424673, leaving just under 3% of the numbers
in the interval. Then I searched for gaps larger than 50206 using a
couple of strong probable prime tests (Miller's test), and a Lucas'
test. The entire process took less than 18 days on a Pentium III 800
loaned from a friend. The search was not so long
but then one should bear in mind that the gap is only just over nine
times the size of the average gap in this region. I checked the gap
a different way but using again probable prime tests and now I am
checking again the composites in the gap with PrimeForm.
I will try to certify that the two probable primes are indeed prime
using Titanix, but for this task I only have available a Pentium II
233 computer and so it can take weeks, barring power failures,
computer crashes, and so on ... any help with this would be
appreciated! Of course, in the extremely unlikely case that some of
these probable primes turned out to be composite, then the found gap
would be even larger.
I do not know where to find gapper, but your two programs (Cramer01 and Cramer02) seem very fast. If I understand them correctly, Cramer01 takes a starting point and finds progressively larger gaps. Cramer02 takes an upper limit on number size and randomly looks below that limit for ever bigger gaps. I need to find all gaps larger than a certain given size within a range of numbers. Is there a Cramer03 that can do this?
P.S. What is Rp?