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