Algorithm for the Collatz Conjecture.

**Is the Collatz conjecture true?**

*“We must know. We will know!”* — **David Hilbert**.

**Remark**: If n = 1, the algorithm terminates all computations.

**Crucial Idea:** **R ^{+}** is a dense set.

**Remark**: **R ^{+}** is the set of all positive real numbers.

We claim

over the Collatz sequence of positive odd integers, from to where the index, t, is the number of trials it takes the Collatz sequence of odd integers to converge to one.

**Remark**: There are no *infinite ( nontrivial) cycles* of any length ( and as ) in the Collatz sequence since the index, t, is clearly finite in equation one.

In addition, our claim is fundamentally based on the distribution of *maximum* divisors, , for *finite set*s of *consecutive* positive even integers (e.g. {2, 4, 6, …, }) where 2 is the minimum positive even integer and where is the maximum positive even integer belonging to those sets…

The positive real number, r, is determined by *the algorithm* for the Collatz conjecture. And therefore, its computation is generally complicated because we cannot easily compute the maximum positive even integer, , in the Collatz sequence (orbit) for many odd positive integers, .

**Question**: For any odd positive integer, , can we approximate the maximum positive even integer, , in the Collatz sequence (orbit)?

The variable, k, is determined by the maximum positive even number, , in the Collatz sequence: .

**Can we refute our analysis?**

We assume maximum divisors, , for any positive even integer.

**Remark**: The Collatz conjecture is true for all positive integers, .

For any positive odd integer, , we have

where .

where .

where .

…

where since and since .

**Question:** Is possible? **No! **

Why? Hint: .

Therefore,

Thus, or

Moreover, where .

We conclude since is a positive *rational* number and since there are no infinite (nontrivial) cycles in any Collatz sequence. Hence .

**Important Remark**: **The Collatz Conjecture is true!** 🙂

**Example**: If we let , then we compute , , and .

Therefore, .

**Dave’s Conjecture**: **O**(t) or for some real number, , such that either or .

**Example**: If we have for some , then as ,

.

Therefore, for infinitely many positive odd integers, .

**Remark**: For our example, we compute {5, 21, 85, 341, …, , …} for all positive integers, .

**Questions**: What are the values (convergent) for ?

**Example**: We compute for infinitely many positive odd integers, .

And therefore, where

{7281, 29125, 116501, 466005,…, , …} for all positive integers, .

**Remark**: For our example, we assume .

**Old Example**: If we let , then we compute , , and .

Therefore, . However,

for infinitely many positive odd integers, .

And therefore, where

{57, 229, 917, 3669,…, , …} for all positive integers, .

**Remark**: For our old example, we assume .

For , we compute the following Collatz sequence where :

172

86

**#1: 43**

130

**#2: 65**

196

98

**#3: 49**

148

74

**#4: 37**

112

56

28

14

**#5: 7**

22

**#6: 11**

34

**#7: 17**

52

26

**#8: 13**

40

20

10

**#9: 5**

16

4

2

**#10: 1**

For , we compute the following Collatz sequence where :

2752

1376

688

344

172

86

**#1: 43**

130

**#2: 65**

196

98

**#3: 49**

148

74

**#4: 37**

112

56

28

14

**#5: 7**

22

**#6: 11**

34

**#7: 17**

52

26

**#8: 13**

40

20

10

**#9: 5**

16

8

4

2

**#10: 1**

**Remark**: The common Collatz sequence of positive odd integers for our old example is

**{43, 65, 49, 37, 7, 11, 17, 13, 5, 1}**.

**Final Remark **(Hmm): The Problem (Collatz Conjecture) has ≡ (mod ) solutions for a given index, t. *The solutions are not unique for a given index, t*. We assume .

**Important Note:**

All odd integers > 1 are either in the sequence, 5, 9, 13, 17, …, 4n +1 or in the sequence, 3, 7, 11, 15, …, 4n – 1 where n > 0.

*******The End of Our Brief Analysis of the Collatz Conjecture *******

**Reference Link**:

Wolfram Cloud (Mathematica Online);

The function, **nvalue[t]**, computes a random positive odd integer, (initially), for a small index, (number of trials for a Collatz sequence of odd integers to converge to one from the **computed** ).

**Source Code**:

**nvalue**[t_] := (

tt = t;

icnt = 0;

n = 2 * RandomInteger[{10, 50000}] + 1;

While[icnt != tt,

While[n != 1,

If[icnt == 0, nstart = n];

n = 3n + 1;

While[EvenQ[n], n = n/2];

icnt = icnt + 1;

If[icnt > tt, n = nstart + 2 *RandomInteger[{1,10}]]

If[icnt > tt, icnt = 0]]];

Return[{tt, nstart}])

__________________________________________________________

**Some** **Examples**:

** nvalue[2]** computes for

**t = 2**a random value, ;

** nvalue[10]** computes for

**t = 10**a random value, ;

** nvalue[5]** computes for

**t = 5**a random value, .

** nvalue[6]** computes for

**t = 6**a random value, .

** nvalue[4]** computes for

**t = 9**a random value, .

** nvalue[1]** computes for

**t = 1**a random value, .

** nvalue[41]** computes for

**t = 41**a random value, .

** nvalue[75]** computes for

**t = 75**a random value, .

** nvalue[75]** computes for

**t = 75**a random value, .

** nvalue[150]** computes for

**t = 150**a random value, .

** nvalue[150]** computes for

**t = 150**a random value, .

** nvalue[200]** computes for

**t = 200**a random value, .

** nvalue[211]** computes for

**t = 211**a random value, .

** nvalue[250]** computes for

**t = 250**a random value, .

** nvalue[300]** computes for

**t = 300**a random value, .

** nvalue[351]** computes for

**t = 351**a random value, .

** nvalue[351]** computes for

**t = 351**a random value, .

** nvalue[408]** computes for

**t = 408**a random value, .

**…**

**Relevant Reference Links**:

**Wolfram Alpha Computation (r _{3})**;

**Collatz Conjecture Calculator****;**

**Our_Response_to_4.2_A_probabilistic_heuristic;**

**The Collatz Equation that supports the Collatz Conjecture;**

**What i****s the importance of the Collatz conjecture?**

**An Analysis of the Collatz Conjecture**;

**THE 3x + 1 PROBLEM: AN OVERVIEW.**

_{“Counting and ordering stuff (objects, sets, numbers, spaces, etc.) are fundamental.”}

**P.S.** **FIGHT SEXISM AND RACISM IN THE SCIENCES INCLUDING MATHEMATICS!** **THANK YOU!**

**Oops! The Proceedings of the London Mathematical Society rejected the paper, “ A Brief Analysis of the Collatz Conjecture“**,

**for publication! Why?**

We are very confident our work is valid, and we suspect our work was rejected because of* political reasons*… It does happen (ostracism, blacklisting, injustice, etc.). But we are also very grateful that Lord GOD is our greatest protector, greatest provider, and our greatest redeemer. Amen!

**More Links: **

**Two Important Properties of Convergent Collatz Sequences**.

Dave.

#### We need your monetary support to complete our current work.

Please make a one-time, monthly, or yearly donation.

#### Please make a monthly donation.

#### Please make a yearly donation

**Choose an amount**:

**Or enter a custom amount**:

**Your contribution is greatly appreciated. Thank you!**

**Your contribution is greatly appreciated. Thank you!**

**Your contribution is greatly appreciated. Thank you!**

This is brilliant 👌

It took me several reviews to grasp what was being conveyed here. My mind is completely blown !

LikeLiked by 1 person

Sparky,

Thank you! 🙂

LikeLiked by 1 person

I just added this web site to my rss reader, great stuff. Cannot get enough!

LikeLiked by 1 person

Thank you! 👍

LikeLike

Wow that was unusual. I just wrote an very long comment but after I clicked submit my comment didn’t appear. Grrrr… well I’m not writing all that over again. Anyhow, just wanted to say excellent blog!

LikeLiked by 1 person

Thank you! 👍

LikeLike

Hello I am so happy I found your weblog, I really found you by error, while I was researching on Google for something else, Anyhow I am here now and would just like to say many thanks for a marvelous post and a all round interesting blog (I also love the theme/design), I don’t have time to browse it all at the moment but I have saved it and also included your RSS feeds, so when I have time I will be back to read more, Please do keep up the great job.

LikeLiked by 1 person

Thank you! 👍

LikeLike

Best view i have ever seen !

LikeLiked by 1 person

Thank you! 👍

LikeLike

You should be a part of a contest for one of the greatest websites on the web. I will recommend this blog!

LikeLiked by 1 person

Way cool! Some extremely valid points! I appreciate you penning this article plus the rest of the site is really good.

LikeLiked by 1 person

I was able to find good advice from your articles.

LikeLiked by 1 person

Quote of the Day:

“In regard to the Collatz conjecture, what’s true for one positive odd integer (either 4n + 1 or 4n – 1 where n > 0) is true for all positive odd integers…”

Do you agree? Why?

LikeLiked by 1 person

This blog was… how do you say it? Relevant!! Finally I have found something which helped me. Cheers!

LikeLiked by 1 person