Problems & Puzzles: Problems
Problem 46 . Holes and Crowds-I
As we run up over the integer numbers we may find large regions primes-free ( holes ) or its contrary, particular regions were primes are as close as they can be (crowds) according to divisibility rules.
Holes are more technically named as gaps; while crowds are better described as prime-constellations or - if we add details as how many primes are in the considered constellation - as k-tuples.
Let's talk in this Problem first about holes (gaps). What is a large hole?
There are two kind of answers:
Precisely Paul Leyland, in the page linked above, keeps the 20 top largest holes measured in both tastes: absolute and relative. As a matter of fact the current (June 2003) hole-records are from JosÚ Luis Pardo (20/20) and from Bertil Nyman (15/20), respectively :
As you can see we, are dealing with recent records produced, let's say, just the day before.
One more word must be said before the questions. The absolute hole-records asked are the earlier non-trivial known holes, that is to say, not the known trivial holes associated to the upper region to each n!+1 value.
Q1. Can you produce one record appropriate for each of the Top 20 Leyland's Tables (*)?
Q2. Do you devise a smart approach to beat these records (some rules to select and/or work the regions in order to seek for holes-size-records)?
I don't want to spoil the nice optimistic and fresh expression of D. H. Lehmer that goes like this "Happiness is just around the corner", said about factoring methods and targets. I just want you to have a small glimpse of the order of magnitude of the records asked to break:
So, the corner of the happiness that D. H. Lehmer mention, for sure exists, but to my conviction, this corner should be near of fast computing machines and smart approaches (the least that I can say is that happiness was not around the corner of my home this week, because I couldn't get, after two heart-selections and two overnights, any interesting result for any of the two Tables :-)
After said all that, let's come the talent, the surprises and the happiness of all of you.
[Next week I will post another problem related to the crowds.]
J. K. Andersen sent (March, 3, 2004) the following interesting records:
The gap ends are 43429-digit probable primes with no simple expression.
My earlier gap size 84630 after p1=2^10093-80445 is still the largest known with a simple expression.
Alm and I have found the currently 9th and 10th largest relative size prime gap with prp'ing by the GMP library. The best has absolute size 7868 and relative size 30.0400 between two 114-digit proven primes: p2 = p1+7868, where p1= 561192545511605501847804031458486579170180721885050519341\
All gaps with larger relative size only have 16 or 17 digits.
We have also found around 20000 (twenty thousand) records for Thomas R. Nicely's site.
Q2 The main strategy of my program is briefly explained athttp://hjem.get2net.dk/jka/math/primegaps/megagap.htm
The most important part is using modular equations to select which numbers in a tested interval are divisible by each small prime, to limit the "overlap" when several small primes divide the same number.
J. K. Andersen wrote (4/6/04):
One more new from J.K.A:
Robert Smith wrote (2016-09-16)
Thank you Robert!