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-   -   LLT numbers, linkd with Mersenne and Fermat numbers (https://www.mersenneforum.org/showthread.php?t=4073)

T.Rex 2005-05-06 14:07

LLT numbers, linkd with Mersenne and Fermat numbers
 
Hi,
I've derived from the Lucas-Lehmer Test a new (??) kind of numbers, that I called LLT numbers. They are described in this short (2.5 pages) paper: [URL=http://tony.reix.free.fr/Mersenne/PropertiesOfLLTNumbers.pdf]LLT numbers[/URL] .
These numbers show interesting numerical relationships with Mersenne and Fermat prime numbers, without any proof yet.

First, I'm surprised it is so easy to create such a kind of numbers that have so close relationships with Mersenne and Fermat numbers. Is there a law saying that playing with prime (Fermat and Mersenne) numbers always lead to nice properties ? :smile:

Second, these numbers may provide interesting primality tests for Fermat and Mersenne numbers (once the properties are proven ...); though they clearly do not improve existing LLT and P├ępin's tests :sad: .

Does someone have hints for proving these properties ? :wink:

Regards,

Tony

Orgasmic Troll 2005-05-06 16:33

watch out for scathing replies, you're definitely abusing terminology here.

T.Rex 2005-05-06 19:42

[QUOTE=TravisT]watch out for scathing replies, you're definitely abusing terminology here.[/QUOTE] Hi TravisT, what's wrong with my paper ? I'm playing with numbers. I did not say I've discovered a magic new method for proving primality of any number. I've just defined and studied the numerical properties of a kind of numbers and noticed some interesting possible properties that need proofs. Can you help me fixing the terminology problems you've noticed ? Can you help providing proofs ?
Thanks,
Tony

Orgasmic Troll 2005-05-06 20:51

[QUOTE=T.Rex]Hi TravisT, what's wrong with my paper ? I'm playing with numbers. I did not say I've discovered a magic new method for proving primality of any number. I've just defined and studied the numerical properties of a kind of numbers and noticed some interesting possible properties that need proofs. Can you help me fixing the terminology problems you've noticed ? Can you help providing proofs ?
Thanks,
Tony[/QUOTE]

taking the "coefficients" of a "function" seems to be a meaningless statement and caught me off guard when I read it. You're taking the coefficients of the polynomials. Since you're never using L as a function, why word it such? In other words, you're never passing a value to L.

I would talk about a set of polynomials P[sub]n[/sub] where P[sub]0[/sub] = x and P[sub]n[/sub] = P[sub]n-1[/sub][sup]2[/sup]-2 where n > 0, I'm no expert, so I may be abusing notation as well.

I haven't had time to look at more than a few of the conjectures you've posed, the first few seem like they can be proven (or disproven) without too much effort

T.Rex 2005-05-07 08:25

Function vs Polynomial
 
You are perfectly right: I should use polynomial rather than function !
I've fixed the mistakes and produced a [URL=http://tony.reix.free.fr/Mersenne/PropertiesOfLLTNumbers.pdf]new version[/URL] .
Seems polynomial x^2-3 has also interesting properties.

So, is there a miracle ? or are these properties an evident consequence of some well-known theorem I'm not aware of ?

Thanks for your comments !

Tony


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