- Apr 30, 2005Hi all,

I wanted to keep these messages together and I add another at the bottom. I

will let you think about how they relate. I no doubt (2nd, no! 3rd, no!) "n-th"

a "Big congratulations on an impressive record" also. ;-)

--- primenumbers@yahoogroups.com wrote:

>

Replace x with a^2+b^2, and y with c^2+d^2 then you have this

> There are 3 messages in this issue.

>

> Topics in this digest:

>

> 1. Re: new simult. prime record with 2058 digits

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

> 2. Re: new simult. prime record with 2058 digits

> From: Gary Chaffey <garychaffey2@...>

> 3. Re: x^y - y^x

> From: "Mark Underwood" <mark.underwood@...>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 1

> Date: Fri, 29 Apr 2005 16:56:56 +0200

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

> Subject: Re: new simult. prime record with 2058 digits

>

> Norman Luhn wrote:

>

> > The name of the big 4-quadruplet is :

> >

> > 4104082046.4800#+5651 {+0,+2,+6,+8}

>

> Big congratulations on an impressive record.

>

> k=4 is the most varied simultaneous record in recent years.

> Since the latest quadruplet, it has been CC 1st kind, CC 2nd kind, BiTwin,

> CPAP,

> CC 2nd kind again:

> http://hjem.get2net.dk/jka/math/simultprime.htm#history

>

> --

> Jens Kruse Andersen

>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 2

> Date: Fri, 29 Apr 2005 16:41:13 +0100 (BST)

> From: Gary Chaffey <garychaffey2@...>

> Subject: Re: new simult. prime record with 2058 digits

>

> I would like to re-itterate Jens congratulations...As a searcher of various

> forms of simultaneous primes I know how hard it must have been to find this.

> Gary

>

> Jens Kruse Andersen <jens.k.a@...> wrote:

> Norman Luhn wrote:

>

> > The name of the big 4-quadruplet is :

> >

> > 4104082046.4800#+5651 {+0,+2,+6,+8}

>

> Big congratulations on an impressive record.

>

> k=4 is the most varied simultaneous record in recent years.

> Since the latest quadruplet, it has been CC 1st kind, CC 2nd kind, BiTwin,

> CPAP,

> CC 2nd kind again:

> http://hjem.get2net.dk/jka/math/simultprime.htm#history

>

> --

> Jens Kruse Andersen

>

>

>

> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com

> The Prime Pages : http://www.primepages.org/

>

>

>

>

>

> ---------------------------------

> Yahoo! Groups Links

>

> To visit your group on the web, go to:

> http://groups.yahoo.com/group/primenumbers/

>

> To unsubscribe from this group, send an email to:

> primenumbers-unsubscribe@yahoogroups.com

>

> Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

>

>

> Send instant messages to your online friends http://uk.messenger.yahoo.com

>

> [Non-text portions of this message have been removed]

>

>

>

> ________________________________________________________________________

> ________________________________________________________________________

>

> Message: 3

> Date: Fri, 29 Apr 2005 20:12:12 -0000

> From: "Mark Underwood" <mark.underwood@...>

> Subject: Re: x^y - y^x

>

> --- In primenumbers@yahoogroups.com, D�cio Luiz Gazzoni Filho

> <decio@d...> wrote:

> > Of note, divisibility of x^y - y^x by p means that x^y == y^x mod

> p. Surely

> > this equation must have interesting properties?

> >

>

> Sure. Of course, if x and y share a factor of p then x^y - y^x does.

> That's why we make sure x and y are relatively prime.

>

> But did we know that x-1 and y-1 must also be relatively prime? It is

> easy to show that if x-1 and y-1 share a factor of p, then x^y - y^x

> does!

>

> Futhermore, the even number -1 must also be relatively prime to the

> other number + 1. It is easy to show that if the even number - 1

> shares a factor p with the other (odd) number + 1, then x^y - y^n has

> the same factor p as well!

>

>

> In a related fashion with Paul's equation of x^y + y^x, not only must

> x and y be relatively prime, but x+1 and y+1 must be relatively

> prime. If x+1 and y+1 share a common factor of p, then so does x^y +

> y^x !

>

> Furthermore, the even number + 1 must be relatively prime to the

> other number -1. If they share a common factor of p, then so does x^y

> + y^x !

>

>

> When we count up all the qualifying x,y pairs up to 500 for x^y - y^x

> and for x^y + y^x (with Paul's equation, x and y's one less than a

> prime are disqualified), we find that about 2.2 times more pairs

> qualify for your expression than for Paul's. This is starting to

> better approach the prime count difference for yours and Paul's

> expressions.

>

>

> Mark

>

http://mathworld.wolfram.com/SquareNumber.html

Equation (19).

See equations (20) and (21) for error with s={-3, -1, 1, 3} as the

"r_0=3^2-s^3 bases". r_0 is the orgin error radius r away from O the true

orgin.

Use equation (7) for an simple proof of Wiles-FLT.

Use equation (33) for proof of a Twin between each prime squared. Or in other

words a prime between n^2 and (n+1)^2.

One might note on the page the 5x^2+/-4 = y^2 iff too. These are the equations

required for twins along with this congruence y +/- 5 == 0 mod (y+6)(y+8).

which is the same as (y+1)^2 == 0 mod (y)(y+2). From a twin one can find "zeros

of admissibility" for other primes.

Another note is that a 3,4,5 and/or 5,12,13 triangles can be used as a logic

sandwich. Remember how close a prime is from a multiple of these numbers. This

is similar to a Dubber Chain of Pythagorean triangles.

The twin Goldbach failure is between 2^6-1 = 7*3^2 (the Primitive Prime factor

exception by Ribenboim's book) and 4900=70^2=2^2*5^*7^2 (Cannonball Problem)is

a "

This proves Goldbach's and Dickson's with the same induction that Erdos used

with Choquet Theory (with different starting and ending points as fitting for

each k-tuple). http://mathworld.wolfram.com/ChoquetTheory.html and

http://mathforum.org/library/drmath/view/51505.html

Thanks to Mark for the the "failure" which is required for Wiles-FLT success

(otherwise there would not be "enough" composites).

>>

John W. Nicholson

>> In addition, I've been looking at the number of prime twins in

>> successively higher, non intersecting intervals from

>>

>> n to n^(1 + 2 / n^(1/e))

>>

>> and the number of prime triplets in intervals from

>>

>> n to n^(1 + 3 / n^(1/e))

>>

>> They have also appeared to only increase, but I have not tested very

>> much.

>

>

>Of course, I have just found a counterexample in the prime triplets.

>A slight modification is in order. :o

>

>Mark

Sorry if this does not make sense, let the questions start.

__________________________________________________

Do You Yahoo!?

Tired of spam? Yahoo! Mail has the best spam protection around

http://mail.yahoo.com - Next post in topic >>