- 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@...

To: primenumbers@yahoogroups.com

Sent: Sun, 6 May 2007 5:14 PM

Subject: [PrimeNumbers] Known prime gaps

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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[Non-text portions of this message have been removed] - --- SWagler@... wrote:
> All,

Prime gaps are far from smoothly distributed.

>

> 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?

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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http://mail.yahoo.com > > Years ago I plotted a frequency distribution of prime gaps from 2 to some

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

> > 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.

All the best,

Andrey

[Non-text portions of this message have been removed]- --- Andrey Kulsha <Andrey_601@...> wrote:
> > > Years ago I plotted a frequency distribution of prime gaps from 2 to some

Except it doesn't venture into the '30 becomes small' region. I can't remember

> > > 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

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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http://mail.yahoo.com - Phil Carmody wrote:
>> A good illustration: http://ieeta.pt/~tos/gaps.html

They are the first two hits on

>

> 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!

http://www.google.com/search?hl=en&q=%22jumping+champion%22

--

Jens Kruse Andersen