That time of year thou mayst in me behold When yellow leaves, or none, or few, do hang Upon those boughs which shake against the cold, Bare ruined choirs, where late the sweet birds sang. In me thou see’st the twilight of such day As after sunset fadeth in the west; Which by and by black night doth take away, Death’s second self, that seals up all in rest. In me thou see’st the glowing of such fire, That on the ashes of his youth doth lie, As the deathbed whereon it must expire, Consumed with that which it was nourished by. This thou perceiv’st, which makes thy love more strong, To love that well which thou must leave ere long.

“In July 1915, Albert Einstein paid a visit to the University of Göttingen (Germany) on the invitation of the mathematician David 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)…”

_{“… In short the Collatz problem is simple enough that anyone can understand it, and yet relates not just to number theory (as described in other answers) but to issues of decidability, chaos, and the foundations of mathematics and of computation. That’s about as good as it gets for a problem even a small child can understand.” — Matt.}

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 addition, our claim is fundamentally based on the distribution of maximum divisors, , for the set of all positive even integers…

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 any odd positive integer, .

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 .

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 .

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 a ≡ mod solution! We assume .