In the formulas for a strange algebraic number in the linked paper, such as X := 1 + sqrt(2*4^(2^13) + 2*sqrt((4^2)^2^13) + 1)) / 2^(14) , if we instead omit the “+1” in the inner square-root sign we would get: 1 + sqrt(2*2*4^(2^13)) / 2^(2^14) = 1 + 2 * 2^(2^13) / 2^(2^14) = 1 + 2^(1 - 2^13) which is very very very close to X. I would think this should go a long way towards explaining the base-2 bit pattern —Dan On Mar 17, 2014, at 6:42 PM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello,
I have been working on a simple idea that lead to something interesting. Some of you may know that the iteration Z(n+1)= Z(n)^2 + c for some simple c will lead to an algebraic number, in this case, many times to a 4'th degree algebraic number.
This is quite puzzling for many reasons, first the convergence to a constant when it happens is geometric, but ??, since we know that the Mandelbrot set is self-similar , what if we take a series of very close rationals (the c's in the equation) , can it be that these numbers would also show some patterns in their binary ? or decimal expansion ?
We can construct a series of rational values and from that get their algebraic equivalent when the formula Z(n+1) = Z(n)^2+c , does converge and then what...,
The question is naive, there is no connection between these two things, but still I was very puzzled by the idea.
So , as usual, since I can't completely understand this phenomena, at the least I made some tables of it, a lot of them, and then, I looked at these numbers, in binary in search of a pattern. By looking at those algebraic numbers I saw, or had the intuition that something is going on with some valuesand found something.
Some algebraic numbers of degree 4 have a very definite and persistent pattern in their binary expansion, the pattern does not go to infinity BUT for some values, it can extend to the first 1000 billion binary digits. , much more than the ordinary approximation. These crazy numbers are chaotic in their continued fractiontoo, not onlythat, the log and the exp is also crazy.
see for yourself :
http://www.plouffe.fr/simon/On%20a%20strange%20class%20of%20algebraic%20numb...
This pattern is astonishing. The one presented on page 1 of the article goes to at least the first 270 million bits,
the numbers are f(n) = 1+1/4*(2*4^n+2*(16^n+1)^(1/2))^(1/2)/(2^n)^2, if n=8192, it gives a pattern that goes for 270 million bits, and for n=1048576, to 1000 billion digits.
a very good question now : can this be generated with a cellular automata ?
I submitted the article to the arxiv site.
As far as I know, this is not known, I searched for some insights with the work of Douady and Hubbard, nothing found, interesting but not definite, maybe it needs a closer look,
Best regards,
Simon Plouffe
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