- --- In primenumbers@yahoogroups.com, Sebastian Martin <sebi_sebi@...>

wrote:>

Along that vein, here's a more general observation:

> Conjecture:

>

> For all prime p>=19

>

> we have

>

> p=3q+2r q and r are both primes.

>

>

> Sincerely:

>

> Sebastián Martín Ruiz

>

there are primes r,q such that

1*r + 2*q produces every prime > 5

2*r + 3*q produces every prime > 17 (as per Sebastian's observation)

3*r + 4*q produces every prime > 31

4*r + 5*q produces every prime > 337

5*r + 6*q produces every prime > 191

6*r + 7*q produces every prime > 421

7*r + 8*q produces every prime > 2711 (hmmm)

8*r + 9*q produces every prime > 881

9*r + 10*q produces every prime > 811

10*r + 11*q produces every prime > 1979.

Just think, somewhere, in some galaxy, a proof for this type of thing

has already been produced. :o

Mark

. - --- Sebastian Martin wrote:
> Conjecture:

It's not new:

>

> For all prime p>=19

>

> we have

>

> p=3q+2r q and r are both primes.

>

>

> Sincerely:

>

> Sebastián Martín Ruiz

http://www.primepuzzles.net/conjectures/conj_047.htm

http://primes.utm.edu/curios/page.php?short=19

Patrick Capelle - I have noticed that all factors of a cyclotomic number Phi(n,b) (that is

the n-th cyclotomic polynomial computed in the point b) are either a

divisor of n or congruent to 1 modulo n. For example:

Phi(13,15) = 139013933454241 = 53 * 157483 * 16655159

All 3 factors are congruent to 1 modulo 13.

Phi(20,12) = 427016305 = 5 * 85403261

5 is a divisor of 20 and 85403261 is congruent to 1 modulo 20.

So I have a few questions:

- is this a general pattern, or is it just another instance of the "law

of small numbers"?

- if so, what is the smallest known counterexample?

- if not, can anyone point me to a (possibly online) demonstration?

Thank you for your interest.

Bernardo Boncompagni

________________________________________________

"When the missionaries arrived, the Africans had

the land and the missionaries had the bible.

They taught how to pray with our eyes closed.

When we opened them, they had the land and we

had the bible"

Jomo Kenyatta

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http://visualtaxa.redgolpe.com

________________________________________________ - --- On Wed, 7/30/08, Bernardo Boncompagni <RedGolpe@...> wrote:
> I have noticed that all factors of a cyclotomic number

There's a well-known equivalent proof for divisors of Fermat Numbers (with an extra stage at the end that you don't need). It's worth understanding that, and then trying to adapt it to arbitrary cyclotomic numbers.

> Phi(n,b) (that is

> the n-th cyclotomic polynomial computed in the point b) are

> either a

> divisor of n or congruent to 1 modulo n. For example:

>

> Phi(13,15) = 139013933454241 = 53 * 157483 * 16655159

> All 3 factors are congruent to 1 modulo 13.

>

> Phi(20,12) = 427016305 = 5 * 85403261

> 5 is a divisor of 20 and 85403261 is congruent to 1 modulo

> 20.

>

> So I have a few questions:

> - is this a general pattern, or is it just another instance

> of the "law

> of small numbers"?

> - if so, what is the smallest known counterexample?

> - if not, can anyone point me to a (possibly online)

> demonstration?

>

> Thank you for your interest.

Phil