- Aargh! I misunderstood the example and posted complete claptrap. My

apologies.

To make amends: all Fermat numbers pass the 2-PRP test. Many composite

Fermat numbers N are known and all are divisible only by integers larger

that 7. In every case, N is a power of 2, by definition, and so not

divisible by 3, 5, 7.

Your conjecture is still false.

Paul

On Mon, 2010-02-01 at 14:34 +0000, Paul Leyland wrote:

> If you used your elderly computer to search for known results rather

> than for computations you would have discovered

> http://www.research.att.com/~njas/sequences/A055550 and, in particular,

> the entry 264239

>

> 264239 = 139 * 1901

> 264238 = 2 * 13 * 10163

>

>

> Paul

>

>

> On Mon, 2010-02-01 at 06:18 -0800, Bill Bouris wrote:

> >

> > Group,

> > I believe that if 'N' passes the 2-PRP test that either 3, 5, or 7

> > divides 'N-1', one or

> > all of them, or 'N' will be divisible by 3, 5, or 7 and be composite.

> > I could only go

> > so far with my outdated computer and UBasic. Can anyone find a

> > counter-example ???

> > Bill

> >

> >

> >

> >

> > - forwarded...

----- Forwarded Message ----

From: Bill Bouris <leavemsg1@...>

To: Paul Leyland <paul@...>

Sent: Mon, February 1, 2010 8:42:15 AM

Subject: Re: {Spam?} [PrimeNumbers] the 357 rule

not a complete craptrap... I was going to say that if it were false, that a simple

3-PRP would expose those numbers that did not have that property.

----- Original Message ----

From: Paul Leyland <paul@...>

To: Bill Bouris <leavemsg1@...>

Cc: pgroup <primenumbers@yahoogroups.com>

Sent: Mon, February 1, 2010 8:39:10 AM

Subject: Re: {Spam?} [PrimeNumbers] the 357 rule

Aargh! I misunderstood the example and posted complete claptrap. My

apologies.

To make amends: all Fermat numbers pass the 2-PRP test. Many composite

Fermat numbers N are known and all are divisible only by integers larger

that 7. In every case, N is a power of 2, by definition, and so not

divisible by 3, 5, 7.

Your conjecture is still false.

Paul

On Mon, 2010-02-01 at 14:34 +0000, Paul Leyland wrote:

> If you used your elderly computer to search for known results rather

> than for computations you would have discovered

> http://www.research.att.com/~njas/sequences/A055550 and, in particular,

> the entry 264239

>

> 264239 = 139 * 1901

> 264238 = 2 * 13 * 10163

>

>

> Paul

>

>

> On Mon, 2010-02-01 at 06:18 -0800, Bill Bouris wrote:

> >

> > Group,

> > I believe that if 'N' passes the 2-PRP test that either 3, 5, or 7

> > divides 'N-1', one or

> > all of them, or 'N' will be divisible by 3, 5, or 7 and be composite.

> > I could only go

> > so far with my outdated computer and UBasic. Can anyone find a

> > counter-example ???

> > Bill

> >

> >

> >

> >

> > - --- In primenumbers@yahoogroups.com,

Bill Bouris <leavemsg1@...> wrote:

> I was going to say that if it were false

n = 1097343303233 is not prime

> that a simple 3-PRP would expose those

> numbers that did not have that property

n is 2-PRP

n is 3-PRP

n is coprime to 3*5*7

n-1 is coprime to 3*5*7

There are 95 such counterexamples in

http://physics.open.ac.uk/~dbroadhu/cert/crxptrxp.gp

the largest being n = 972137210279970593

David