Processing math: 100%
     

Exercises - The Collatz Conjecture

  1. Suppose we define f(x) in the following way, for positive integers x:
    f(x)={x/2 if x is even3x+1 if x is odd

    1. Construct the sequence of values a0,a1,a2,... so that a0=n and ai+1=f(ai). Verify that this sequence ultimately gets stuck in the loop 4, 2, 1, 4, 2, 1, ... for the following values of n:

      1. n=21
      2. n=13
      3. n=31
    2. Find the sequence described in part (a) for other values of n. What do you notice?

    3. Suppose we use L(n) to denote the position where 1 first occurs in the sequence so generated and starting with n. Show that if n=8k+4, then L(n)=L(n+1).

    4. Show that if n=128k+28, then L(n)=L(n+1)=L(n+2).

     
    1. See below...

      21 → 64 → 32 → 16 → 8 → 4 → 2 → 1 ...

      13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 ...

      31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → 137 → 412 → 206 → 103 → 310 → 155 → 466 → 233 → 700 → 350 → 175 → 526 → 263 → 790 → 395 → 1186 → 593 → 1780 → 890 → 445 → 1336 → 668 → 334 → 167 → 502 → 251 → 754 → 377 → 1132 → 566 → 283 → 850 → 425 → 1276 → 638 → 319 → 958 → 479 → 1438 → 719 → 2158 → 1079 → 3238 → 1619 → 4858 → 2429 → 7288 → 3644 → 1822 → 911 → 2734 → 1367 → 4102 → 2051 → 6154 → 3077 → 9232 → 4616 → 2308 → 1154 → 577 → 1732 → 866 → 433 → 1300 → 650 → 325 → 976 → 488 → 244 → 122 → 61 → 184 → 92 → 46 → 23 → 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 ...

    2. Every number seems to generate a sequence that gets stuck in the loop 4, 2, 1, ...

    3. Consider the first 4 iterations:

      8k+44k+22k+16k+4
      8k+524k+1612k+86k+4

      From there on, the sequences are identical, and thus get stuck in the loop 4,2,1 at the same time.

    4. Consider the first 4 iterations:

      128k+2864k+1432k+796k+22
      128k+29384k+88192k+4496k+22
      128k+3064k+15192k+4696k+23

      Notice, the first two sequences have merged. Now consider the next 7 iterations:

      96k+2248k+11144k+3472k+17
      216k+52108k+2654k+13162k+40

      96k+23288k+70144k+35432k+106
      216k+53648k+160324k+80162k+40

      So now, all three sequences have merged, which implies they will get stuck in the loop 4,2,1 at the same time.