Me again, my wife's out of town so I'm surfing the net too much. Apologies
if you're sick of me.
> b) All (or all but finitely many with an upper bound or maximum quantity
> for the sporadics) Mersenne primes follow some pattern.
> Imagine there turned out to be a link between primes patterns (or Mersenne
> prime patterns) and the Mandelbreot set? It's not out of the question.
That
> thing has interesting additive combinatorics, also doubling patterns,
> Fibonacci sequences, and the like hidden in it.
It's absolutely doubtless there is a link with the Mandelbrot set - and
hopefully I won't fall in to the usual trap of trendy mumbo-jumbo and black
magic that the subject usually yields. It might be doubtful that we could
ever *use* this fact, but who knows, stranger things have happened,
apparently the pretty colors can be generated by charging a metal plate in
the shape of the set itself...
As Paul says, the thing is oozing with combinatorics. Each of its components
has an associated cycle length, in fact, each component has a root point
which, if the iteration is applied n times, the point is mapped back to
itself. Components (or "sprouts") are connected at single "attachment
points", and the cycle length of a child component is a multiple of its
parent. A lot of analysis has been done on the location of these points.
But think of it like this. Suppose you wanted to know how many components
were of each cycle length n. You'd have to find the root points, ie solve
"n'th iterate of x=x", which is an equation of order (you guessed it)
2^(n-1). Factor out the root x=0 (the root of cycle length 1) and the
equation has a Mersenne number degree. Of course, some of these roots are
also roots for factors of n, but you can enumerate them - you'll also end up
running for the Cunningham tables. The strange thing is, since the number of
components of given cycle length grows exponentially, there are not enough
"attachment points" for them all. Hence the familiar "mini-Mandelbrot" sets
have to appear, seemingly in the middle of nowhere (though they are
connected via some very convoluted infinite sequences - the set is, I think,
proven to be a single connected piece).
<Hand-waving>
Prime cycle lengths are interesting, because of the connection points
problem. A prime cycle length *should* produce more "island molecules" than
a composite would. In theory then we could do a primality test; zoom very
closely on a *very* tiny area of the Mandelbrot set (which we could compute
by some iterative root-finding procedure) and have a look. If there's an
"island molecule" there, N is prime, if not, N is composite. Obviously,
mathematical accuracy - and rendering time - is quite an issue if you're
staring so closely at the thing, but it's an intriguing thought experiment.
Sounds radical? Not at all, we've all seen this process 38 times before...
just not in the complex plane:
The Mandelbrot set iteration z -> z^2+c in C
The Lucas test iteration x -> x^2-2 in Z(N).
It's just too much to be a coincidence. A Lucas test is nothing more than a
very thin slice of the Mandelbrot set. Island molecules have a lot in common
with Lucas pseudoprimes.
</Hand-waving>
Chris Nash
Lexington KY
UNITED STATES
=======================================================
Co-discoverer of probably the 8th and 11th largest known primes.
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