Re: [math-fun] Question about base 3/2

25 Sep 2016

      Hello,

  bases 2 and 3 do not mix well in general,

  I am not certain I understand your question, you say in base (3/2) but 
your example uses
base (2/3).

Nevertheless, in base (3/2) the number 1/3 has the following expansion.

  [0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 
1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0,
1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 
0, 0, 1, 0, 0, 1, ...]

  no pattern here.

Also, it is well known that {(3/2)^n}, that is the fractional part of 
(3/2)^n is not
simple in base 2 or 3.

for example : the fractional part of (3/2)^3000 is in binary :

0.0111101110001000001001011111100110010110100101000100100111101001011001110\
100010101110110011111111000110011001101101011100100110111110101000110001\
111100110110100010100001100111000000011000000001000011111101001011010101\
111101101000101001011011000010101110111000110100010100101001001101111010\
010101000001100010111111011100110010111000101110100011001110011110010100\
101111100011011010011010111101100011011101101011010111000010000111101111\
101011001010000011100110001110111110100101100100010110111100100101111001\
100001011100110111010000101110110101000100010010111110000011101011100111\
001001100001110010011100111000010010000111100111111100111101001001001001\
010101101110111000101100000010011100000010110011101000101100111001011011\
100101110001101100011010111110110001001101100001010111100000001100101001\
010010110110001100011010010010101000101000000011101100111100111100011100\
001010011001001101101001010001010000001000111000010000000101001000000010\
010011001000011000001100010101100110000101100110101001011010101100010000\
101010011101001000111100111010001000000000000000000111111101011100000010\
010010100001011010001110110101101000111100000000000111000110011111100110\
101111001011001001101000100111111111001110111010001010011001101001010111\
011110011010100101110000010000011111010001001111010100000110111110011011\
101111000001111100000100101100100011011111011001000000000100101000000011\
111000110011010011010001111000101010010011110001110001000010101000101000\
001110001011010001011011110100010101111010001111100111001101101010001101\
110100001111101010010101001011110000111010010100110110110000000011111101\
001001001011101000100010100010101110000001111101011001011100010011000001\
011100110100110010110001100101000000010010111110111001111100000111110111\
111000001010110011010111110010000101010111010001101000001001000101010111\
011011101001111000000000111000000001110011101011111001000101011110111010\
101001101100111110000001000110001000111110101010111000010011110110110011\
110011011101000000001000011111010101101111100110000110010100100011101110\
101001101110110011100100011001111001111101001010111010000001001100010110\
011101110011100111100101011000000110101000110000000000111110100000010011\
001101011101001100010011110000011101010111001011101010010100010100000110\
011001010110000110000010101000001101011110100010011011111101110000100100\
010110010111011110101000011101001000101000000100010000100011001111111110\
010011101111110010110101100001111011100000001110100010110000101011101000\
101111111011100000101111111010110110101010001100010110000111001010010111\
011110100010110000101010110011101101000100010011010101010110100101001011\
100111100010001101000001111011111000000011110100110111101111100110000000\
110010010010101101111011011110100101000110001110001100011110000101100100\
010001001110110101100111110101101111101100000000000000001100011101001011\
001010100010011000100001111110001001100100101011001001011011001100010000\
110001010111100110010101110100001100001101101000011110001100001100001110\
100011101001001100010001011110110110101111001110110001001000010001000010\
100000000010010000001000101110001110010111000001011100011101001001101100\
000001110001001101110101110000111100000010010100110111000101111101100010\
101000011101010000100110100101111011010001111100010000111101101110110111\
001111111100111011100110011101110101100010100001000010110011100011000010\
110111110010001000010000101011110101001101011111110100111000000100"

as you can see : ...??? I do not recognize any simple pattern.

  I know that some people worked hard to get results in that direction, 
but so far
this is intractable by classical means.

  Best regards,

  Simon Plouffe

Le 2016-09-25 Ã  05:39, James Propp a Ã©crit :
...
Is it possible to write 1/3 as an infinite sum of the form a_1 (2/3)^1 +
a_2 (2/3)^2 + ... where the sequence of nonnegative integer coefficients
a_1, a_2, ... is eventually periodic?
Jim Propp
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