## Re: [PrimeNumbers] Known prime gaps

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• All, Years ago I plotted a frequency distribution of prime gaps from 2 to some small limit and the curve always looked similar to the curve for black body
Message 1 of 11 , May 7, 2007
All,

Years ago I plotted a frequency distribution of prime gaps from 2 to some small limit and the curve always looked similar to the curve for black body radiation. Has anyone done this for limits large or small? Are there theoretical reasons to account for this?

Steve Wagler

-----Original Message-----
From: jens.k.a@...
Sent: Sun, 6 May 2007 5:14 PM

Polignac's conjecture says all even prime gap sizes occur infinitely many
times. So far the only known way to prove existence of a gap size is to find
an occurrence.

Thomas R. Nicely maintains tables of first known occurrence prime gaps at
http://www.trnicely.net/gaps/gaplist.html
For each gap size the smallest known consecutive primes or prp's with that
gap are listed.

Torbjörn Alm has searched first known occurrence gaps for a long time with a
sieve by me, using modular equations to ensure unusually many small factors
in wanted gaps. Small prp tests are made by the GMP library, and large by
PrimeForm/GW.

There is now a proven occurrence of all 10000 even gaps up to 20000.
Marcel Martin's Primo proved the large majority of the 20000 gap ends.

In addition, there is now either a proven or prp occurrence of all even gaps
up to 30000, and currently of 30046 even gaps in total (and 1 odd!).
Torbjörn found the listed occurrence of 21274 of them. Others had previously
found larger primes for some of the gap sizes. It is not recorded who was
the first to find an occurrence of a gap.

The Top-20 Prime Gaps at
http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm lists the best gaps in
different categories.
The merit of the gap from p1 to p2 is defined as (p2-p1)/log p1, where log
p1 is the average gap size in that vicinity.
This year Torbjörn has found the 3 largest known gaps with merit above 20.
The best is a gap of 114554 between 2227-digit primes. The merit is 22.34.

--
Jens Kruse Andersen
___
.

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• ... Yes. Credit for doing something harder but more natural . I recall an old Guinness edition which in addition to the official world record listed fastest
Message 2 of 11 , May 7, 2007
Mike Oakes wrote:
> Sorry, I'm probably being stupid and missing your point, which is: you
> want to give extra credit where /no/ such technique is employed, don't
> you?

Yes. Credit for doing something harder but more "natural".
I recall an old Guinness edition which in addition to the official world
record listed "fastest 100m at sea level" (there is less air resistance in
thin air at altitude).
The other table comment, "Largest gap with proven end points",
is also extra credit.
Finding large merits with no modular technique requires a huge
number of attempts.
Such gaps get "unfair" competition from "artificial" modular constructions.
Without the basic expression listing, they would have no entry for
merit above 20, and until recently no entry for merit above 10.

--
Jens Kruse Andersen
• ... Prime gaps are far from smoothly distributed. Gaps divisible by 3 are more likely than ones not divisible by 3. As 30 becomes small, gaps divisible by 30
Message 3 of 11 , May 8, 2007
--- SWagler@... wrote:
> All,
>
> Years ago I plotted a frequency distribution of prime gaps from 2 to some
> small limit and the curve always looked similar to the curve for black body
> radiation. Has anyone done this for limits large or small? Are there
> theoretical reasons to account for this?

Prime gaps are far from smoothly distributed.
Gaps divisible by 3 are more likely than ones not divisible by 3.
As 30 becomes small, gaps divisible by 30 also become more popular.
As always this can be explained by looking at small primes.

Phil

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• ... A good illustration: http://ieeta.pt/~tos/gaps.html All the best, Andrey [Non-text portions of this message have been removed]
Message 4 of 11 , May 8, 2007
> > Years ago I plotted a frequency distribution of prime gaps from 2 to some
> > small limit and the curve always looked similar to the curve for black body
> > radiation. Has anyone done this for limits large or small? Are there
> > theoretical reasons to account for this?
>
> Prime gaps are far from smoothly distributed.
> Gaps divisible by 3 are more likely than ones not divisible by 3.
> As 30 becomes small, gaps divisible by 30 also become more popular.
> As always this can be explained by looking at small primes.

A good illustration: http://ieeta.pt/~tos/gaps.html

All the best,

Andrey

[Non-text portions of this message have been removed]
• ... Except it doesn t venture into the 30 becomes small region. I can t remember where 30 takes over on from 30 as the most likely gap. I presume that s
Message 5 of 11 , May 8, 2007
--- Andrey Kulsha <Andrey_601@...> wrote:
> > > Years ago I plotted a frequency distribution of prime gaps from 2 to some
> > > small limit and the curve always looked similar to the curve for black
> body
> > > radiation. Has anyone done this for limits large or small? Are there
> > > theoretical reasons to account for this?
> >
> > Prime gaps are far from smoothly distributed.
> > Gaps divisible by 3 are more likely than ones not divisible by 3.
> > As 30 becomes small, gaps divisible by 30 also become more popular.
> > As always this can be explained by looking at small primes.
>
> A good illustration: http://ieeta.pt/~tos/gaps.html

Except it doesn't venture into the '30 becomes small' region. I can't remember
where 30 takes over on from 30 as the most likely gap. I presume that's touched
on somewhere on the prime pages or on mathworld, and if it isn't on both,
something should be done about that!

Phil

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• ... They are the first two hits on http://www.google.com/search?hl=en&q=%22jumping+champion%22 -- Jens Kruse Andersen
Message 6 of 11 , May 8, 2007
Phil Carmody wrote:
>> A good illustration: http://ieeta.pt/~tos/gaps.html
>
> Except it doesn't venture into the '30 becomes small' region. I can't
> remember
> where 30 takes over on from 30 as the most likely gap. I presume that's
> touched
> on somewhere on the prime pages or on mathworld, and if it isn't on both,
> something should be done about that!

They are the first two hits on