It seems to work, but why ? - Page 2

Numbers · 2005-10-12, 16:50

That will teach you for switching your PC off. You should have it running 24/7 with prime95 using it while you are asleep.

T.Rex · 2005-10-13, 21:05

Quote:

Originally Posted by T.Rex

The methods used by Lucas and Lehmer do not apply for F_5 and f_1. Tony

I was wrong.
Based on Lehmer technics and on my own previous work, I think I have a proof for (LLRT ?):

$\large \ \ \ \ M_q \text{ is a prime iff } M_q \ \mid \ S_{q-2} \text{ where: } S_0=5 \ , \ S_{i+1} = 2 S_i^2-1$ .

The proof takes 1 page, based on theorems appearing in Williams' book.

I'll write it in $L_AT_EX$ asap in order to check I did not make any mistake, and then I will provide a link to my web-site, so that everyone can check the proof.

(I really prefer that test rather than the LLT, because it mimics the form of Mersenne numbers: $M_q = 2 (2^x)^2-1$ ! Also, their computing costs are quite identical.)

Tony

T.Rex · 2005-10-14, 21:25

Hi all,

Here is a draft of the proof of a new (AFAIK) primality test for Mersenne numbers, also based (like LLT) on Lucas Sequences and Lehmer theorems. (Should I name it: LLRT ?)

I would appreciate early readers to check it.
Do you think it is worth to propose it for publication in a Mathematic Journal ?

It is close to the old LLT, but it flies through the Mersenne number space in a different way, producing different intermediate numbers and residues.

Stay tuned, I will add some more properties.
Also, I have to fix some minor mistakes in the document I refer to in the paper.

For those who know nothing about Lucas Sequences and about Lucas-Lehmer theorems, just know that proving this theorem is easy, because Lucas and Lehmer have done the most difficult work. Thanks also to HC Williams for his wonderful book: "Edouard Lucas and Primality Testing".
I think the most difficult part was to imagine that there could be a different way for proving primality of Mersenne numbers.

Though providing a new primality proof for Mersenne numbers that has the same cost than LLT seems unuseful at first, I think that it may help the search of Mersenne primes and of Mersenne factors.

Regards,

Tony

R. Gerbicz · 2005-10-14, 23:22

I think you waste your effort.

For Lucas Lehmer test x[0]=10 is also a good starting value, x[i+1]=x[i]^2-2.
See http://primes.utm.edu/mersenne/
Your Lucas sequence for Mersenne numbers is: s[0]=5, s[i+1]=2*s[i]^2-1. It is easy to prove ( by induction ) that s[n]=x[n]/2, so Mq divides s[q-2] if and only if Mq divides x[q-2] ( because Mq is odd) . So it is not a new discovery.

T.Rex · 2005-10-15, 10:38

Quote:

Originally Posted by R. Gerbicz

I think you waste your effort ...

You are perfectly right, and I've been (nearly) completely stupid.

(I thought to check that it produces different numbers, but I forgot to look at the other roots of LLT:
$\large U_0=4 \ ,\ U_{n+1}=U_n(U_n^2-3)$ .)

At least, I've found some mistakes in the second paper.
Is this paper (LLT for Fermat numbers) correct ? or did I make another (big) mistake ?

Thanks for your help !
Tony

2005-10-14, 21:25	#14
T.Rex Feb 2004 France 2²×229 Posts	A proof of a new (AFAIK) Primality test for Mersenne numbers. Hi all, Here is a draft of the proof of a new (AFAIK) primality test for Mersenne numbers, also based (like LLT) on Lucas Sequences and Lehmer theorems. (Should I name it: LLRT ?) I would appreciate early readers to check it. Do you think it is worth to propose it for publication in a Mathematic Journal ? It is close to the old LLT, but it flies through the Mersenne number space in a different way, producing different intermediate numbers and residues. Stay tuned, I will add some more properties. Also, I have to fix some minor mistakes in the document I refer to in the paper. For those who know nothing about Lucas Sequences and about Lucas-Lehmer theorems, just know that proving this theorem is easy, because Lucas and Lehmer have done the most difficult work. Thanks also to HC Williams for his wonderful book: "Edouard Lucas and Primality Testing". I think the most difficult part was to imagine that there could be a different way for proving primality of Mersenne numbers. Though providing a new primality proof for Mersenne numbers that has the same cost than LLT seems unuseful at first, I think that it may help the search of Mersenne primes and of Mersenne factors. Regards, Tony

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2005-10-12, 16:50	#12
Numbers Jun 2005 Near Beetlegeuse 2²×97 Posts	That will teach you for switching your PC off. You should have it running 24/7 with prime95 using it while you are asleep.

2005-10-14, 23:22	#15
R. Gerbicz "Robert Gerbicz" Oct 2005 Hungary 2×743 Posts	I think you waste your effort. For Lucas Lehmer test x[0]=10 is also a good starting value, x[i+1]=x[i]^2-2. See http://primes.utm.edu/mersenne/ Your Lucas sequence for Mersenne numbers is: s[0]=5, s[i+1]=2*s[i]^2-1. It is easy to prove ( by induction ) that s[n]=x[n]/2, so Mq divides s[q-2] if and only if Mq divides x[q-2] ( because Mq is odd) . So it is not a new discovery.