> >I think we should check the math first. I have a sneaky suspicion that looping
> >won't occur in the relevant region (the first 2^n-3 iterations) unless n is
> >composite - which may be interesting, but doesn't help us eliminate Mersenne
> >numbers as candidate primes. But my math is inadequate to prove this 8-(
>
> I think that you mean n-2 iterations, but you may be right.
Yup. Brain failure again. I think I must have been confused with the
idea that the sequnce _must_ recur within 2^n-3 iterations - there
are only 2^n-1 different values of the residual, and 2 of them are
special - see below
> It's hard to
> say, without any evidence, or solid math.
Chris Nash's excellent contribution isn't an actual proof, but it looks
solid enough to make me _extremely_ sceptical about the value
of searching for a recurrence, so far as testing Mersenne numbers
for primality is concerned. If you did find an example, you'd be well
worth your "15 minutes of fame"!
>
> Just a side note, but all l_n values are 2 after n-1 iterations on a mersenne
> prime. Maybe lends some small bit of evidence against that...
Yes but the recurrence _starts_ after n iterations. (n-2)th is 0,
(n-1)th is -2, all subsequent values are +2
Also, if the sequence ever reaches +1 or -1 it immediately
starts recurring, 1^2 - 2 = -1, (-1)^2 - 2 = -1, ...
This is why we can reduce the maximum recurrence period
from 2^n-1 to 2^n-3 since both -1 and +2 have two "precursors".
Regards
Brian Beesley
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