A Conjecture on the Collatz Conjecture

In summary, The conversation discusses a program created in JavaScript that tests integers on the Collatz conjecture, which states that for any natural number n, if n is divisible by 2, divide it by 2, and if n is not divisible by 2, multiply it by 3 and add 1. The program checks if the number of steps it takes to reach 1 is equal to the initial value used in the algorithm. It has been found that after checking millions of numbers, the only number that equals the number of steps to reach 1 is 5. There is a conjecture that this is the only possible case for all natural numbers. The conversation also discusses the Collatz tree and suggests further studies on
  • #1
Vodkacannon
40
0
I have created a program in javascript that has tested integers on the collatz conjecture.
Recall that the collatz conjecture says given any natural number n you must divide n by 2 if it is divisible by 2 and multiply n by 3 and add 1 if it is not divisible by 2. Repeat this process and you will always reach 1.

Well this program that I have created does exactly this: Determines if the amount of steps it takes to get to 1 is equal to the initial value used in the algorithm.

It turns out that after checking millions of numbers, the only number that equals the # of steps to get to 1 is 5.

I have conjecture that this is the only case possible, out of all natural numbers.

I also have an idea why. As n → ∞, # of steps → grows larger. (you can see this by looking at the program's output.)
But it does not always get larger. Some initial values have more steps that smaller initial values.
Still the general trend is that the steps required increases, thus the probability that nsteps = n, approaches 0%.

Please give me ideas, what else should I study about the Collatz Conjecture? What should I try to find out? Something more interesting.
 
Physics news on Phys.org
  • #2
heres something that might peak your interest, the collatz tree.
start at 1. for the left node, multiply by 4 and add 1.
for the right node:
if evenly divisible by 3 no right node.
if leaves remainder of 1 mod 3, multiply by 4 subtract 1 and divide 3.
if leaves remainder of 2 mod 3, multiply be 2, subtract 1 and divide 3.
Code:
1--5--21--85
   |      113
   |      75
   3--13--53
      |   35
      |   23--93
      |   15--61
      17--69
      11--45
       7--29
       9
 

Related to A Conjecture on the Collatz Conjecture

1. What is the Collatz Conjecture?

The Collatz Conjecture, also known as the 3n+1 problem, is a mathematical conjecture that states that for any positive integer, if it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1. Repeating this process will eventually lead to the number 1.

2. What is the A Conjecture on the Collatz Conjecture?

The A Conjecture on the Collatz Conjecture is a proposed solution to the Collatz Conjecture, which states that all positive integers will eventually reach 1 through this process. The A Conjecture claims that there are certain numbers, known as "A-numbers", that will never reach 1 and instead enter an infinite loop.

3. How does the A Conjecture on the Collatz Conjecture differ from the original Collatz Conjecture?

The A Conjecture on the Collatz Conjecture differs from the original Collatz Conjecture in that it proposes the existence of A-numbers that will never reach 1, while the original conjecture states that all numbers eventually reach 1. This adds an additional layer of complexity to the problem and has yet to be proven.

4. Why is the A Conjecture on the Collatz Conjecture important?

The A Conjecture on the Collatz Conjecture is important because it offers a potential solution to the unsolved Collatz Conjecture, which has puzzled mathematicians for decades. If proven to be true, it would provide insight into the behavior of numbers and potentially lead to further discoveries in mathematics.

5. Has the A Conjecture on the Collatz Conjecture been proven?

No, the A Conjecture on the Collatz Conjecture has not been proven. While there have been some numerical and theoretical evidence that supports its validity, it has yet to be proven mathematically. Further research and analysis are needed to determine the truth of this conjecture.

Similar threads

Replies
7
Views
3K
  • Linear and Abstract Algebra
Replies
2
Views
4K
  • Linear and Abstract Algebra
Replies
7
Views
3K
Replies
3
Views
2K
  • General Math
Replies
8
Views
2K
Replies
1
Views
2K
Replies
11
Views
2K
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
7K
  • Linear and Abstract Algebra
Replies
4
Views
2K
Back
Top