## Known prime gaps

Expand Messages
• 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
Message 1 of 11 , May 6, 2007
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
• ... gaps in ... where log ... above 20. ... 22.34. ... Jens: Congratulations on adding yet another /very/ nice page to your site. The only sticking point I
Message 2 of 11 , May 7, 2007
--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
>
> 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: Congratulations on adding yet another /very/ nice page to your
site.

<<
A basic expression is here defined as maximum 25 characters, all
taken from 0123456789+-*/^( ). Primorial and factorial are not
allowed since they can be used to ensure many small factors, and the
idea of the basic expression record is partly to avoid special prime
gap methods.
>>

A rule that excludes the 2 characters ! and # seems weird.

Is not Pierre's
PRP38007 = 50491*(87811#)/6 - 657714
every bit as transparent and explicit as Milton's
PRP14173 = 10^14173 - 51197
?

You would seem to be in danger of being thought to be penalizing
users of the humbler n*p#+k construct viz a viz your own more
sophisticated methods.

Just my 4c :-)

-Mike Oakes
• 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 3 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
___
.

________________________________________________________________________
AOL now offers free email to everyone. Find out more about what's free from AOL at AOL.com.

[Non-text portions of this message have been removed]
• ... Thanks. It s actually an old page originally started by Paul Leyland: http://hjem.get2net.dk/jka/math/primegaps/leylandgaps20.htm The only new page is
Message 4 of 11 , May 7, 2007
Mike Oakes wrote:
> Jens: Congratulations on adding yet another /very/ nice page to your
> site.

Thanks. It's actually an old page originally started by Paul Leyland:
http://hjem.get2net.dk/jka/math/primegaps/leylandgaps20.htm

The only new page is "Ormiston Tuples" (consecutive primes containing the
same digits): http://hjem.get2net.dk/jka/math/ormiston_tuples.htm
The largest known number of such primes is 9:
26460346024426922096587598498580390201381951306930145595901871467050710000
+ (7839, 7893, 7983, 8379, 8397, 8739, 8793, 8937, 8973)

> <<
> A basic expression is here defined as maximum 25 characters, all
> taken from 0123456789+-*/^( ). Primorial and factorial are not
> allowed since they can be used to ensure many small factors, and the
> idea of the basic expression record is partly to avoid special prime
> gap methods.
> >>
>
> A rule that excludes the 2 characters ! and # seems weird.

They are excluded for what they represent: An expression designed to have
specific values (in this case all 0's) modulo a lot of numbers. I want to
show the best gap which reaches merit above 10 or 20 without a
construction that increases the chance of large gaps. It doesn't matter
whether a single character has been defined to express the construction.
I am not considering to permit ! and #, but I might permit other things if
they don't allow modular constructions. As far as I know, nobody has
searched large prime gaps expressed with other operators or functions than
+-*/^#!, so it's not relevant now. I'm not spending time considering a lot
of hypothetical functions which are never used for prime gaps. It's only a
small part of the site.

> Is not Pierre's
> PRP38007 = 50491*(87811#)/6 - 657714
> every bit as transparent and explicit as Milton's
> PRP14173 = 10^14173 - 51197
> ?

Yes, but transparent and explicit are not considerations - and explicit
decimal expansions are on subpages (unlike Nicely's site which has to list
far more gaps).

> You would seem to be in danger of being thought to be penalizing
> users of the humbler n*p#+k construct viz a viz your own more
> sophisticated methods.

Really? The rule for "basic expressions" only applies to two gaps at the
site. None of them are currently by me and I have no plans to search for
replacements. I have been a bit harsh in public on Milton's gap
announcements, so I don't think people will suspect me of doing him a
favour over Pierre.
By the way, Pierre's n*p#/6+k has better gap performance on average (but not
in worst case) than n*p#+k. I don't know whether Pierre was first to notice
this, and I haven't analyzed whether 6 is the optimal divisor to exclude
from the primorial.

--
Jens Kruse Andersen
• ... have ... Yet, is it not true that your own Chinese Remainder Theorem technique is equally designed to do precisely just that ! (I obviously can t sway you,
Message 5 of 11 , May 7, 2007
--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
>
>> A rule that excludes the 2 characters ! and # seems weird.
>
> They are excluded for what they represent: An expression designed to
have
> specific values (in this case all 0's) modulo a lot of numbers.

Yet, is it not true that your own Chinese Remainder Theorem technique
is equally designed to do precisely just that !

(I obviously can't sway you, so I'll shut up now.)

-Mike Oakes
• ... 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? My bad.
Message 6 of 11 , May 7, 2007
--- In primenumbers@yahoogroups.com, "Mike Oakes" <mikeoakes2@...>
wrote:

> Yet, is it not true that your own Chinese Remainder Theorem technique
> is equally designed to do precisely just that !
>

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?

Mike
• ... 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 7 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 8 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

() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com
• ... A good illustration: http://ieeta.pt/~tos/gaps.html All the best, Andrey [Non-text portions of this message have been removed]
Message 9 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 10 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

() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com
• ... They are the first two hits on http://www.google.com/search?hl=en&q=%22jumping+champion%22 -- Jens Kruse Andersen
Message 11 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