Saturday, December 27, 2014

3x+1 and T^n(k)'s evaluation matrix

In previous posts I've been talking about the 3x+1 problem and the fabulous T(x) function that satisfies

T^n(k2^n + r) = k3^j + T^n(r)

If you let 0 <= r < 2^n, and evaluate T^m(x) for all the resulting values, and for 1 < m <= n, you get a so-called evaluation matrix.  In this evaluation matrix, such as the one below,
  • 16k + 0    8k + 0    4k + 0    2k + 0    1k + 0
  • 16k + 1    24k + 2    12k + 1    18k + 2    9k + 1
  • 16k + 2    8k + 1    12k + 2    6k + 1    9k + 2
  • 16k + 3    24k + 5    36k + 8    18k + 4    9k + 2
  • 16k + 4    8k + 2    4k + 1    6k + 2    3k + 1
  • 16k + 5    24k + 8    12k + 4    6k + 2    3k + 1
  • 16k + 6    8k + 3    12k + 5    18k + 8    9k + 4
  • 16k + 7    24k + 11    36k + 17    54k + 26    27k + 13
  • 16k + 8    8k + 4    4k + 2    2k + 1    3k + 2
  • 16k + 9    24k + 14    12k + 7    18k + 11    27k + 17
  • 16k + 10    8k + 5    12k + 8    6k + 4    3k + 2
  • 16k + 11    24k + 17    36k + 26    18k + 13    27k + 20
  • 16k + 12    8k + 6    4k + 3    6k + 5    9k + 8
  • 16k + 13    24k + 20    12k + 10    6k + 5    9k + 8
  • 16k + 14    8k + 7    12k + 11    18k + 17    27k + 26
  • 16k + 15    24k + 23    36k + 35    54k + 53    81k + 80
there are various k 2^a 3^b + r expressions.  I just proved that all such expressions satisfy 0 <= r < 2^a 3^b.  In other words, T^m(r) is the remainder of dividing T^m(x) by the corresponding 2^a 3^b.

2 comments:

Anonymous said...

Hi Andrés!

It's a semi unrelated question. Well; it is not a question, but a suggestion for a new post (:

The challenge:

x^x=1000
x=??

I passed calculus 1, 2 and 3. I used to know how logarithm works, but with this, don't even know how to start. Any advice??

TIA!

Andrés said...

> I used to know how logarithm works ... any advice??

How about picking up a book? If you don't know which book, how about learning how to find books that can help? I've found this "round about" way of attacking problems wins in the long run because it enables your future success... you learn how to fish yourself.