Collatz's Conjecture

General Idea:

Collatz’s conjecture was a problem posed by L. Collatz in 1937. Let a0 be an integer. The Collatz problem asks if iterating:

(Wolfram Alpha)

always returns to one for any a0. Here are a few examples of Hailstone sequences or Collatz’s sequences:

a0 = 17:

17,52,26,13,40,20,10,5,16,8,4,2,1

a0 = 51:

51,156,78,39,118,59,178,89,268,134,67,202,101,304,152,76,38,19,58,29,88,44,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1

a0 = 23:

23,70,35,106,53,160,80,40,20,10,5,16,8,4,2,1

Collatz’s problem has been tested by computers and the result is that for any a0 ≤ 19(258), it always returned to one. Here are two graphs from Wolfram Alpha that show how many steps it takes to reach one for a0 = n:

These graphs are very interesting. We can see that in the first graph almost no numbers exceed 100 steps and in the second graph almost no numbers exceed 200 steps. Most of the 1-200 numbers are in the 0-25 steps section, and the numbers between 1 and 2000 are bundled in the 1-50 steps section, and there is also a small crowd in the 100 and 150 steps section for the 1-2000 section. So, why is this so? Well, go home and study some Calculus for a couple of years, then you should be able to figure it out.

Are you back? Yeah, you were probably too lazy for that. Anyways, Terras modified the Collatz problem and his version is shown below:

This is a conjecture, and, it has not been proven. You can generate some hailstone sequences here: https://plus.maths.org/content/mathematical-mysteries-hailstone-sequences. You can also code your own function, remember, here is the problem:

Let a0 be an integer. The Collatz problem asks if iterating:

(Wolfram Alpha)

always returns to one for any ao.

More In-Depth Study of the modified Terras problem:

Terras proved that the set of integers  has a limiting asymptotic density. Let's call it F(k), such that if  is the number of n such that n is less than or equal to x and , then the limit

exists.  Also, as k heads toward infinity, F(k) heads toward 1. Finally, for all k larger than or equal to one:

where

(Lagarias 1985).

For more, visit http://mathworld.wolfram.com/CollatzProblem.html.

(Wolfram Alpha)

This is a very interesting problem that has been bickered about by mathematicians for nearly 100 years. The problem basically is asking, then, that with any integer to start with, if the number mod 2 is 1, multiply by 3 and add 1, and if the number mod 2 is zero, divide the number by two. Then, the Collatz problem says that this integer will eventually reach a power of two - because that is the only way for a number to reach one in this problem. Then, it will go into the endless loop of 4, 2, 1, 4, 2, 1, 4 and so on.

Try to calculate some sequences by hand. See if you find any patterns. You can also run some computer programs to make graphs and calculate for you. Anyway, the possibilities are endless. At least for now.