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)

> The only "sticking point" I had when reading it was this:-

> <<

> 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