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

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• ... From: Jens Kruse Andersen ... Well, then what I was asking for, the smallest primes with biggest merit, is a subset of that list: 2
Message 1 of 3 , Dec 31, 2005
----- Original Message -----
From: "Jens Kruse Andersen" <jens.k.a@...>

>First known occurrence prime gaps are at
>http://www.trnicely.net/gaps/gaplist.html
>For every gap size, it lists the smallest known primes or prp's with exactly
>that gap.

Well, then what I was asking for, the smallest primes with biggest merit, is a subset of
that list:

2 -->1.44
3 --> 1.82
7 --> 2.06
113 --> 2.96
1129 --> 3.13
1327 --> 4.73
19609 --> 5.26
31397 --> 6.95
155921 --> 7.19
360653 --> 7.50
370261 --> 8.70
1257201 --> 9.35
2010733 --> 10.20
17051707 --> 10.81
20831323 --> 12.46
191912783 --> 13.00
436273009 --> 14.18
2300942549 --> 14.84
4302407359 --> 15.96
10726904659 --> 16.54
25056082087 --> 19.04
304599508537 --> 19.44
461690510011 --> 19.81
1346294310749 --> 20.84
1408695493609 --> 21.02
1968188556461 --> 21.27
2614941710599 --> 22.80
13829048559701 --> 23.66
19581334192423 --> 25.03
218209405436543 --> 27.44
1693182318746371 --> 32.28
55350776431903243 --> 31.07
80873624627234849 --> 31.34

If I sieved well, this is more or less the current state of the real smallest primes with
biggest merits.
Is it in the OEIS? (I'll check soon).
Are there any heuristics to predict the size of the next merit and prime?
Are there upper bounds that can never be trespassed, for a given size?

Thank you for all the useful info you sent in your email.

Jose
• ... From: Jens Kruse Andersen ... Yes, that was a residue from the comparations I did to check I had sieved well, the list should stop
Message 2 of 3 , Dec 31, 2005
----- 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.

In my list I had:

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
• From: Jose Ramón Brox Date: 12/31/05 13:10:50 To: Prime Numbers Subject: [PrimeNumbers] RE Prime Gaps (Was Prime gaps of 364,188 ) ... From: Jens Kruse
Message 3 of 3 , Jan 2, 2006
From: Jose Ramón Brox
Date: 12/31/05 13:10:50
To: Prime Numbers
Subject: [PrimeNumbers] RE Prime Gaps (Was "Prime gaps of 364,188")

----- Original Message -----
From: "Jens Kruse Andersen" <jens.k.a@...>

>First known occurrence prime gaps are at
>http://www.trnicely.net/gaps/gaplist.html
>For every gap size, it lists the smallest known primes or prp's with
exactly
>that gap.

Well, then what I was asking for, the smallest primes with biggest merit, is
a subset of
that list:

2 -->1.44
3 --> 1.82
7 --> 2.06
113 --> 2.96
1129 --> 3.13
1327 --> 4.73
19609 --> 5.26
31397 --> 6.95
155921 --> 7.19

From Kermit
kermit@...

I looked at the first few primes in the list mod 2, 3, 5, 7, 11, 13, 17

in hopes of seeing a simple pattern. I did not see it.

Mod2Mod 3Mod 5Mod 7Mod 11Mod 13Mod 17
2357111317
20222222
31033333
71120777
11312313911
112911427117
13271124711
196091142758
3139712223215
155921121371214

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