Yes, I forgot to add : those numbers are easily approximable but : consider that there is a grey area where a number is quite recognisable and cannot be completely be expressible by rationals or irrationals, which is the case here. The approximation is quite good considering continued fractions but that does not explain the hundred of millions of bits that we can see instantly. respectfuly, Simon Plouffe 2014-03-18 15:58 GMT+01:00 Simon Plouffe <simon.plouffe@gmail.com>:
Hello,
The formula I have found is not a discovery, what is somewhat surprising is the order of approximation. Consider only the formula sqrt(sqrt(2^(2^n)+1)) is simple enough and not related to Mandelbrot, the surprising fact is that for small values of n, the pattern is extending for billions of binary digits from which a good half of them are 100 % predictable.
Best regards, Simon Plouffe