Hi,
I've put on my web-site 2 documents providing the proof of 3
primality tests for Fermat numbers, based on P. Ribenboims'
and HC Williams' books, and on Lucas Sequences.
(It is possible that these tests are not new ... but I did not
find them in books and on the web.)
Nevertheless, I would appreciate if someone could check if the
proofs is correct, or not.
The first 2 tests are interesting since they show that "fixed
points" may occur in the S_i sequence for different values of
n in F_n=2^2^n+1, here: S_{2^n-n-2} which does not reduce a
lot the computation.
Being able to build a proof for fixed points would help since
I have found a more useful fixed point with another Lucas
Sequence: S_{2^{n-2}-2} , which may reduce the time needed for
proving primality by 25 % .
The third test is interesting since it shows that the LLT
(S_{i+1} = S_i^2 - 2) can be used for Fermat numbers with a
different S_0 : 5 .
http://tony.reix.free.fr/Mersenne/PrimalityTest1FermatNumbers.pdf
http://tony.reix.free.fr/Mersenne/PrimalityTest2FermatNumbers.pdf
My idea is that there are still important things to discover
within Lucas Sequences that could reduce the time needed for
proving the primality of Fermat numbers. Maybe it could also
help for Mersenne numbers.
Regards,
Tony
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