LLT numbers, linkd with Mersenne and Fermat numbers
Hi,
I've derived from the LucasLehmer 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 Pépin's tests :sad: . Does someone have hints for proving these properties ? :wink: Regards, Tony 
watch out for scathing replies, you're definitely abusing terminology here.

[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 
[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]n1[/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 
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^23 has also interesting properties. So, is there a miracle ? or are these properties an evident consequence of some wellknown theorem I'm not aware of ? Thanks for your comments ! Tony 
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