Hello, each image of rank k is : the k'th position of n*n*phi n is from 1 to 16000000 disregarding the decimal point. A pattern appears for every k, The construction of the page uses When n*n*n is used only the first few k appears to have a pattern, the image is quite scrambled already when k = 12. The same thing would appear if we construct an image with the squares starting at position 1 of n*n , we have, 1, 4, 9 , 1, 2, 3, 4, 6, 8, 1, 1, 1, 1, 1, 2, ... : seq A002993. Best regards, Simon Plouffe Le 2017-01-07 à 15:26, Gareth McCaughan a écrit :
On 07/01/2017 01:03, Simon Plouffe wrote:
I made one experiment with n*n*phi, phi being the golden ratio.
I use a precision of let's say 1000 digits and n = 1 to 16000000.
If we look at the square grid made of the numbers (without the decimal point), there are no clear patterns in the lines of digits BUT there is a definite pattern in vertical.
for each k (from 1 to 1000) I constructed 1 image made out of the first 16 000 000 digits , making one image of 4000 x 4000. ... These digits are in base 10, so I used a colourization with blue tones : light blue = 0 , darker blue = 9.
The thing is : there is a pattern for every k. Very definite pattern.
I'm not sure I understand. Are n and k the same thing? Does each image show the decimal digits of n^2 phi for some n? (That doesn't seem like it can be right, because you say that if you use n^3 instead of n^2 then "it does scramble after a few digits" but some numbers are both squares and cubes. But maybe the patterns are only visible for smallish n or something?)
Another thing I don't think I understand: it looks to me as if the patterns are two-dimensional rather than being "in the lines of digits" or "in vertical".
And another, this time in the results rather than in the description: It seems that we get big blocks in which one kind of pattern persists quite stably, and quite abrupt transitions between one block and another. I wouldn't have predicted that sort of phase transition. ... Hmm, it looks to me as if these transitions, when they are visible, occur at fairly consistent positions in the decimal expansion. Maybe they correspond to "interesting" numbers of decimal places or something.