Hi,

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

in

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.

Jose Luis.