## 17468RE Prime Gaps (Was "Prime gaps of 364,188")

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• Dec 31, 2005
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----- Original Message -----
From: "Jens Kruse Andersen" <jens.k.a@...>

Jose Ramón Brox wrote:

>> 1693182318746371 --> 32.28
>> 55350776431903243 --> 31.07
>> 80873624627234849 --> 31.34

>The two last merits are below Nyman's lucky 32.28 which was record
>for 6 years. I haven't checked the rest.

Yes, that was a residue from the comparations I did to check I had sieved well, the list
should stop at 32.28.

A typewritting typo (I changed a 3 for a 2).
Two forgotten merits.
The two residual last merits.

I know because it is in the OEIS and I compared it, it is A111870. By the way, I think it
has an error, so I reported it (4652353 appears before 2010733, and moreover, the merit of
the second is bigger than the merit of the first, so the first one shouldn't be on the
list - unless I didn't see it well or Nicely's has an error himself).

>The Prime Pages has heuristic-based conjectures about large prime gaps:
>http://primes.utm.edu/notes/gaps.html
>It doesn't mention merit directly but can be translated to merits.
>It says that in 1931 Westzynthius proved there are arbitrarily large merits.

Yes, I know about his theorem, but I can't find a proof in the net (I think it should be
one in H&W or something like that, but I don't have the book). Could be that you know
where to find it?

>Bertrand's postulate (proved by Chebyshev) says there is a prime between
>n and 2n.
>This gives a weak upper bound on merits for primes below a given size.

Of course! If n is prime then L <= n-1, m = L / log (n) <= (n-1) / log(n), and indeed it's
a week bound because, for example, I think it's proven that we can find two primes between
n and 2n, one of the form 4n-1 and other of the form 4n+1 - besides there are much better
boundings for primes.

Jose
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