“In July 1915, Albert Einstein paid a visit to the University of GĂ¶ttingen (Germany) on the invitation of the mathematicianDavid Hilbert (1862-1943). It was a fruitful encounter for both men that continued over the following months with an intense scientific correspondence. Einstein described that period as the most exhausting and stimulating of his entire life, the result of which was a series of studies and articles, authored by one or the other scientist, in which they formulated the equations of the gravitational field of the General Theory of Relativity (GTR)…”

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 sets 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! đź™‚

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 â‰ˇ (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, â€¦, 4n +1 or in the sequence, 3, 7, 11, 15, â€¦, 4n â€“ 1 where n > 0.

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

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 ).

_{“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 ofÂ 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!