I don’t know if this is even a proper question, but:

The LL test with starting values of 4 generates an accompanying sequence of Heronian triangles, as in: [13,14,15],[193,194,195],[37633,37634,37635]

The second to last and final LL residual sometimes have an accompanying Pythagorean triple (depending on the Lehmer symbol), as in: [15,112,113],[511,130560,130561],[1023,523264,523265]

Does this suggest some geometric analogy underlying the test that involves triangles?

Sorry if this is a silly question!

Andy

]]>Wainting for someone to have a look… ]]>

Consider the field $F=F_{q^2}$ and recall that there is a homomorphism $n: F^\times \to F_q$ given by $N(x)=x\cdot x^q$, the norm. Let $N$ denote the kernel of this homomorphism, i.e. the subgroup of elements of norm 1.

Then $N$ has order $p+1$ and is cyclic as it is a finite subgroup of the multiplicative group of a field.

The magic of the Lucas-Lehmer test is that $2+\sqrt{3}$ is a generator for this group, regardless of which Mersenne prime we consider.

But let me stress that the fact that there is such an element (of norm 1) follows from general theory. ]]>

See: http://www.primenumbers.net/prptop/prptop.php?page=1#haut

Regards,

Tony Reix

Look at: http://tony.reix.free.fr/Mersenne/SummaryOfThe3Conjectures.pdf

and: http://trex58.wordpress.com/math2matiques/

for more information.

Tony ]]>

Thanks for the proof. I’ll study it and see if it can help me.

I’m one of the crazy guys who contribute to the GIMPS project, since ages.

I’ve contributed to checking the last 6 Mersenne primes, on a big Itanium2 machine, with the wonderful Glucas program written by Guillermo Ballester Valor.

I also spent a lot of time studying the LLT, reading E. Lucas books, HC Williams book about E. Lucas work, Ribenboim famous book about primes, and so on.

I’ve also found some properties of Mersenne numbers and of the LLT, with proof of myself or by other people. Mq=(8x)^2-(3qy)^2. As an example for LLT, since it is not well known that the LLT can be used to prove that a Fermat number is prime (a number N such that N-1 is easily factorized), I wrote such a proof, based on Ribenboim technic and on Williams’ book.

So, though I am an “amateur”, I have some knowledge of the LLT test.

The LLT test makes use of the tree that appears in the DiGraph under x^2-2 modulo a Mersenne prime (see Shallit & Vasiga for details). This DiGraph also has many cycles, of lengh dividing q-1. I found the formula that gives the number of cycles of lengh L, under x^2 and x^2-2 modulo a Mersenne prime. Someone else built the proof.

I also have conjectured a test that makes use of such a cycle of length q-1 (with a fix seed) instead of the tree (with seed 4). The main difference with the basic LLT test is that the last step of the iteration, instead of leading to 0 (mod Mq), leads to the seed of the iteration: S_0. Thanks to some help of Mr Williams, I’ve been able to prove that, if Mq is prime, then the property holds. But I failed finding a way to prove the reverse…

In short, the x^2-2 builds trees and cycles modulo a number. If the number is prime, there is symmetry. If the number is composite, then trees and cycles are not symmetric. My idea is that cycles can be used as the tree is used for proving that a number is prime.

This would not be very interesting to have another test for Mersenne numbers, since it would not be faster than LLT.

But Anton Vrba built a conjecture about using the same technique for a primality proof for Fermat 2^2^n+1 and Wagstaff (2^q+1)/3 numbers. And he provided a proof for the first (easy) part: a Prime number HAS this property.

About Wagstaff numbers, proving the conjecture would lead to a brand new primality test as efficient as the LLT is for Mersenne numbers. That would be the first VERY efficient test for a number not N-1 nor N+1. But Anton failed to prove the reverse… Some Mathematician contributing to the GIMPS Math forum think that, though it should be possible for the Mersenne, it does not seem possible to prove the conjecture for Wagstaffs. However, the DiGraph under x^2-2 modulo a Wagstaff prime only has cycles. No tree.

About Fermat numbers (tree and cycles), I guess (just a guess…) that it could be possible to build a test that could be used to prove that a Fermat number is NOT prime, much faster than running the full Pépin’s test. This could be possible since the Digraph under x^2-2 modulo a Fermat prime generates cycles of length dividing 2^n-1. I expect that a Fermat number that does not verify such a (much shorter) test could not be a prime. Just an idea… but it could help to know what is the status of F33.

So, there is here an area of research. Simply studying the properties of the cyles and trees for the Mersenne, Fermat and Wagstaff numbers under x^2 or x^2-2 would be very interesting, and useful for further reseach. It could be the subject of a study for a Math student (too hard for me…).

Succeeding in proving the conjectures would be a great success !

If you are interested by this subject, just contact me at

tony . reix @ laposte . net .

Regards,

Tony

I was working on the Mersenne primes and it appears there is “some pattern” in all the known Mersenne primes. I have verified it for the 46 known Mersenne Primes and it holds.

I am not sure if its something “easily provable” or we do have something that has gone unnoticed before.

Need to know if you would be interested to see the pattern/conjecture and comment on the same.

Thanks in advance.

Tarandeep Singh

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