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.

The traditional holes (gaps) record-keeper pages are these maintained by Thomas R. Nicely and by Paul Leyland; while for crowds (k-tuples) the best place I know is the Tony Forbes's one.

Let's talk in this Problem first about holes (gaps). What is a large hole?

There are two kind of answers:

a) a large hole as an absolute large hole or simply the distance between two successive primes: (p2-p1)

b) a large hole as a relative large hole respect to the expected size of the holes in the region we are talking about: (p2-p1)/ln(p2). It happens that the expected size of a hole in the neighborhood of p2 is ln(p2), so...

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 José 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:

1) How many digits has the number after which we get the trivial gap 233833; that is to say how many digits has 233833!+1? I will save you the calculations (Stirling's formula courtesy), but this number has something more than a million of digits, and the relative size of a gap like this, supposing primes at the extremes of this gap, would be something like 0.1 (against the 17.28 of the Gomez Pardo's record)

2) Probably you remember the result gotten by J. K. Andersen last week for the Puzzle 223. He found a remarkable gap between the primes p1=2^10093-80445 and p2= 2^10093+4185. This is a gap of absolute size, p2-p1= 84630 < 117930 and relative size (p2-p1)/ln(p2) = 84630/ln(2^10093+4185) = 84630/6996 = 12.09 <28.60. As you can see this remarkable gap can not displace the smallest (the 20th) record consigned in each of the two Tables.

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.

J. K. Andersen wrote (4/6/04):