- Hello,

I recently found the following statements :

Let L(k) be the Lucas sequence L(0)=2, L(1)=1, L(k+2) = L(k+1) + L(k)

Let M(n, p) = (n+1)^p - n^p for n >= 1 and p prime >= 3 (M(n,p) can

be viewed as a generalization of Mersenne's number).

1/ Let n >= 1 such that n mod 5 is 0, 2 or 4

Then :

M(n, p) is prime if and only if M(n, p) divides L((n+1)^p) - L(n^p

+ 1)

2/ Let n >= 1 such that n mod 5 is 1 or 3 and let p prime >= 3 such

that p mod 4 is 1

Then :

M(n, p) is prime if and only if M(n, p) divides L((n+1)^p) - L(n^p

+ 1)

It follows an efficient primality test for M(n, p) just like the well

known Lucas-Lehmer test for Mersenne's numbers.

I would like your feedback on this. Is it a known result ?

Best regards,

J-L Charton. - --- j_chrtn <j_chrtn@...> wrote:
> Hello,

Welcome to the list.

> I recently found the following statements :

Yes, one that's been used by several before you.

>

> Let L(k) be the Lucas sequence L(0)=2, L(1)=1, L(k+2) = L(k+1) + L(k)

>

> Let M(n, p) = (n+1)^p - n^p for n >= 1 and p prime >= 3 (M(n,p) can

> be viewed as a generalization of Mersenne's number).

> 1/ Let n >= 1 such that n mod 5 is 0, 2 or 4

I'm not familiar with it. I suspect that it's not a proven result as a

> Then :

>

> M(n, p) is prime if and only if M(n, p) divides L((n+1)^p) - L(n^p

> + 1)

>

>

> 2/ Let n >= 1 such that n mod 5 is 1 or 3 and let p prime >= 3 such

> that p mod 4 is 1

> Then :

>

> M(n, p) is prime if and only if M(n, p) divides L((n+1)^p) - L(n^p

> + 1)

>

>

> It follows an efficient primality test for M(n, p) just like the well

> known Lucas-Lehmer test for Mersenne's numbers.

>

> I would like your feedback on this. Is it a known result ?

primality test. It might well be a compositeness test (Probable Primality

test). The way primiality proofs almost always work is that they limit the

size and form of divisors of the number in question (e.g. if you prove they're

all greater than the square root of the number, then you're sorted). I don't

see how satisfying your criterion alone says anything about the size of

divisors.

Aside - who is recording these primes, I don't think it's one of Jens' or

Steven's forms, is it?

Phil

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http://mail.yahoo.com - Hi Phil,

--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...>

wrote:>

(k)

> --- j_chrtn <j_chrtn@...> wrote:

> > Hello,

>

> Welcome to the list.

>

> > I recently found the following statements :

> >

> > Let L(k) be the Lucas sequence L(0)=2, L(1)=1, L(k+2) = L(k+1) + L

> >

can

> > Let M(n, p) = (n+1)^p - n^p for n >= 1 and p prime >= 3 (M(n,p)

> > be viewed as a generalization of Mersenne's number).

(n^p

>

> Yes, one that's been used by several before you.

>

> > 1/ Let n >= 1 such that n mod 5 is 0, 2 or 4

> > Then :

> >

> > M(n, p) is prime if and only if M(n, p) divides L((n+1)^p) - L

> > + 1)

such

> >

> >

> > 2/ Let n >= 1 such that n mod 5 is 1 or 3 and let p prime >= 3

> > that p mod 4 is 1

(n^p

> > Then :

> >

> > M(n, p) is prime if and only if M(n, p) divides L((n+1)^p) - L

> > + 1)

well

> >

> >

> > It follows an efficient primality test for M(n, p) just like the

> > known Lucas-Lehmer test for Mersenne's numbers.

as a

> >

> > I would like your feedback on this. Is it a known result ?

>

> I'm not familiar with it. I suspect that it's not a proven result

> primality test. It might well be a compositeness test (Probable

Primality

> test). The way primiality proofs almost always work is that they

limit the

> size and form of divisors of the number in question (e.g. if you

prove they're

> all greater than the square root of the number, then you're

sorted). I don't

> see how satisfying your criterion alone says anything about the

size of

> divisors.

Well, you're right: at this time, I have unfortunately not been able

to prove these assertions. However, as far as I know Lucas-Lehmer

test is based on the rank of apparition of a given prime P in the

Fibonacci or Lehmer extension sequences and the relationships between

these sequences and their companion Lucas or Lehmer extension

sequences. I suspect that my two statements above may be proved the

same way ...

>

Jens' or

> Aside - who is recording these primes, I don't think it's one of

> Steven's forms, is it?

I don't know if someone officially records this form of primes. I

just know that we (Mike Oakes, Predrag Minovic, myself and maybe

other people) have recorded many PRP of this form to Henri Lifchitz

probable primes page.

The largest known PRP of this form (which is my current record) is

10^282493-9^282493.

J-L.