Chebytchev - Page 2

maxal · 2006-02-10, 23:09

I have added some info about initial values for LLT to MersenneWiKi page but I believe that page contains a wrong statement about so-called "universal" initial values. Let me repeat here what I said at the discussion page:

I believe that the following statement is wrong:

Samuel Gebre-Egziabher has shown that there are only 3 Universal Initial Values than can be used for all Mersenne numbers: 4, 10 and $\frac{2}{3} \,\eq \,\frac{2^{q}+1}{3}$

There are many other "universal" initial values, for example, given by the sequence A018844. Actually, all initial values in the new section (that I added) are essentially "universal" unless we take them modulo a particular M_q. So I think the incorrect statement above should be removed and the term "universal" should be dropped. Please correct me if I'm wrong.

T.Rex · 2006-02-11, 22:32

Hi maxal,

Not sure I'm wrong. Maybe I've not provided all the needed definitions and theorems.

I've taken this interesting information from a "Thesis presented to The Division of Mathematics and Natural Sciences of Reed College" in May 2002 by Robert Thompson. This appears in Chapter 2 "Initial Values for Lucas-Lehmer Sequences". I only have a paper copy.

Definition 2.1 says:
"Let e be a rational number and {e_j} be the Lucas-Lehmer sequence with initial value e_1=e . We shall call e a "universal" initial value for Mersenne numbers 2^p-1 if $e_{p-1} \ \equiv \ 0 \ \pmod{2^p-1} \ \text{<==>} \ 2^p -1 \text{ is prime }$ for all odd primes p."
Then the author continues and shows that only 4, 10 and 2/3 are rational numbers that can be used as initial value for all primes p .

The proof appeared first in a paper of Mr Gebre-Egziabher I do not have. He is now finishing his thesis in Europe.

I mean:
- 4 leads to the serie: 4, 52, 724, 10084, ...
- 10 leads to the serie: 10, 970, 95050, ...
- 2/3 leads to the serie: 2/3, -46/27, 482/243, 3022/2187, ...

So the idea is: a) we knew 4 and 10 ; b) the thesis shows that 2/3 has the same role as 4 and 10 have.

Tony

maxal · 2006-02-12, 19:51

Quote:

Originally Posted by T.Rex

"Let e be a rational number and {e_j} be the Lucas-Lehmer sequence with initial value e_1=e . We shall call e a "universal" initial value for Mersenne numbers 2^p-1 if $e_{p-1} \ \equiv \ 0 \ \pmod{2^p-1} \ \text{<==>} \ 2^p -1 \text{ is prime }$ for all odd primes p."
Then the author continues and shows that only 4, 10 and 2/3 are rational numbers that can be used as initial value for all primes p .

The trouble is in the "only" statement. Yes, 4, 10, and 2/3 can be used as the initial value for LLT. But there are many others!

Quote:

Originally Posted by T.Rex

The proof appeared first in a paper of Mr Gebre-Egziabher I do not have.

I would really like to see the proof since I believe it's flawed.

Quote:

Originally Posted by T.Rex

I mean:
- 4 leads to the serie: 4, 52, 724, 10084, ...
- 10 leads to the serie: 10, 970, 95050, ...
- 2/3 leads to the serie: 2/3, -46/27, 482/243, 3022/2187, ...

So the idea is: a) we knew 4 and 10 ; b) the thesis shows that 2/3 has the same role as 4 and 10 have.

No objections unless you claim that 4, 10, and 2/3 are the only initial values. What's wrong with 52, for example?

2006-02-10, 23:09	#12
maxal Feb 2005 2²×3²×7 Posts	I have added some info about initial values for LLT to MersenneWiKi page but I believe that page contains a wrong statement about so-called "universal" initial values. Let me repeat here what I said at the discussion page: I believe that the following statement is wrong: Samuel Gebre-Egziabher has shown that there are only 3 Universal Initial Values than can be used for all Mersenne numbers: 4, 10 and $\frac{2}{3} \,\eq \,\frac{2^{q}+1}{3}$ There are many other "universal" initial values, for example, given by the sequence A018844. Actually, all initial values in the new section (that I added) are essentially "universal" unless we take them modulo a particular M_q. So I think the incorrect statement above should be removed and the term "universal" should be dropped. Please correct me if I'm wrong.

2006-02-11, 22:32	#13
T.Rex Feb 2004 France 2²·229 Posts	Hi maxal, Not sure I'm wrong. Maybe I've not provided all the needed definitions and theorems. I've taken this interesting information from a "Thesis presented to The Division of Mathematics and Natural Sciences of Reed College" in May 2002 by Robert Thompson. This appears in Chapter 2 "Initial Values for Lucas-Lehmer Sequences". I only have a paper copy. Definition 2.1 says: "Let e be a rational number and {e_j} be the Lucas-Lehmer sequence with initial value e_1=e . We shall call e a "universal" initial value for Mersenne numbers 2^p-1 if $e_{p-1} \ \equiv \ 0 \ \pmod{2^p-1} \ \text{<==>} \ 2^p -1 \text{ is prime }$ for all odd primes p." Then the author continues and shows that only 4, 10 and 2/3 are rational numbers that can be used as initial value for all primes p . The proof appeared first in a paper of Mr Gebre-Egziabher I do not have. He is now finishing his thesis in Europe. I mean: - 4 leads to the serie: 4, 52, 724, 10084, ... - 10 leads to the serie: 10, 970, 95050, ... - 2/3 leads to the serie: 2/3, -46/27, 482/243, 3022/2187, ... So the idea is: a) we knew 4 and 10 ; b) the thesis shows that 2/3 has the same role as 4 and 10 have. Tony