RE Prime GAP of 364,188
----- Original Message -----
From: "Jens Kruse Andersen" <jens.k.a@...>
> I'm not sure what counts as closed formula.
I was thinking in a formula that is valid for every N without changing it's parameters,
that is, all of them are in terms of N: you write the formula once and we can use it for
whenever N we desire.
>Primorials are better than factorials for gap producing.
Yes, I was just thinking about this :-)
> There are better methods to choose which numbers have small factors.
> Such methods were used for the 2 known Megagaps:
I will attach here an extract from your link:
"n# is the product of all primes <= n. The bounding primes were found on the form p1 =
279893*100343#+c-81498 and p2 = p1+1001548, where c is a 43423-digit number with no simple
expression. c was chosen modulo all primes up to 100343, to ensure unusually many numbers
(994427) with a factor <= 100343 in an interval of 10^6 following k*100343#+c for any k.
The found gap and that interval do not line up completely but that could not have been
Decimal expansions of c and p1 are in a text file.
k was chosen after sieving of 1 million intervals to 10^8 and some of these to 2^32, to
ensure many of the 5572 remaining unfactored numbers had a factor between 100343 and
Indeed a very intelligent approach: I was wondering how all that records could be arrived
at, and now I know. Are there any other approaches, or yours is the main used one?
Unfortunately, this procedure isn't a closed formula in the sense I stated above. If we
change N for anyone bigger than 10^6, we need to recompute k and c and probably the number
in the place of 100343, without having a clear binding relation between them and N.
A nice field, this of the prime gaps.
Do you know if the smallest gaps with biggest merit are being collected somewhere? I don't
refer to the biggest known, but to the real ones, starting with the first prime gap
(between 3 and 5) and finishing in the frontier of too-hard-to-compute prime gaps.