Two Important Properties of Convergent Collatz Sequences

[Guest]

Property I: From every convergent Collatz sequence, we can form a bounded Collatz set, [tex]C_{even }[/tex], of positive even integers.

Property II: The union of all possible bounded Collatz sets, [tex]C_{even }[/tex], of positive even integers, is the set, E, of all positive even integers.

What are the proofs of properties, I and II?

Hints: The proofs are trivial, but their implications may not be trivial in regards to the truth of the Collatz Conjecture. Please ponder the algorithm for the Collatz Conjecture.

Important Question: Does there exists an exceptional set, [tex]\Tau[/tex], of positive even integers, [tex]\tau[/tex], that violates the Collatz Conjecture?

Hints: Please review the proof of Collatz Conjecture at the link below and refer to the algorithm for the Collatz Conjecture.

And try to find/construct at least one positive even integer that violates the Collatz Conjecture or prove that it exists.

Good luck!

Remarks: You will be famous if you can completely prove or disprove the Collatz Conjecture for all positive even integers. Any half measures are unacceptable! For example, the recent questionable proof of the Collatz Conjecture for almost all positive integers by Terence Tao is unacceptable!

Relevant Reference Links:,

'Collatz Conjecture',

https://en.wikipedia.org/wiki/Collatz_conjecture;

'Proof of Collatz Conjecture'

https://www.math10.com/forum/viewtopic.php?f=63&t=1485&start=20.

[Guest]

FYI: 'ALMOST ALL ORBITS OF THE COLLATZ MAP ATTAIN ALMOST BOUNDED VALUES' by TERENCE TAO,

https://arxiv.org/pdf/1909.03562.pdf.

[Guest]

FYI: 'ALMOST ALL ORBITS OF THE COLLATZ MAP ATTAIN ALMOST BOUNDED VALUES' by TERENCE TAO,

https://arxiv.org/pdf/1909.03562.pdf.

What a questionable and misleading paper!

Relevant Reference Link:

'Mathematics must maintain its purity.'

https://www.math10.com/forum/viewtopic.php?f=63&t=8270&sid=f6d09017f1148f0f5f55ca211519a23f.

[Guest]

FYI: "A Brief Analysis of the Collatz Conjecture",

https://theory-of-energy.org/2020/09/17/a-brief-analysis-of-the-collatz-conjecture/.

[Guest]

FYI: "A Brief Analysis of the Collatz Conjecture",

https://theory-of-energy.org/2020/09/17/a-brief-analysis-of-the-collatz-conjecture/.

Oops! The paper, "A Brief Analysis of the Collatz Conjecture", was rejected as unacceptable by the Proceedings of the London Mathematical Society. Please refer to the attached file below.

Hmm. Why?

[Guest]

FYI: "A Brief Analysis of the Collatz Conjecture",

https://theory-of-energy.org/2020/09/17/a-brief-analysis-of-the-collatz-conjecture/.

Oops! The paper, "A Brief Analysis of the Collatz Conjecture", was rejected as unacceptable by the Proceedings of the London Mathematical Society. Please refer to the attached file below.

Hmm. Why?

We are confident our work is valid, and we suspect our work was rejected because of political reasons... It happens.

Dave.

[Guest]

FYI: Mathematica code for finding some starting values (positive odd integers), [tex]21 \le n_{0} < 100,000[/tex] (approximately), for some indices,[tex]1 \le t \le 10[/tex].

1. Wolfram Cloud (Mathematica Online), https://www.wolframcloud.com/; (Please login/register for a new/free basic account...)

2. Published Source Code Link, https://www.wolframcloud.com/obj/openmind123omega/Published/collatzcalc.nb.

Remark: The function, nvalue[t], computes a random positive odd integer, 21 \le n_{0} \le 100,000 (approximately), for a small index, 1 \le t \le 10 (number of trials for a Collatz sequence of odd integers to converge to one from the computed n_{0}).

Source Code:

nvalue[t_] := (
tt= t;
icnt = 0;
n = RandomInteger[{21, 100000}];
If[EvenQ[n], n = n + 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, n_{0} = 54, 613;

nvalue[10] computes for t = 10 a random value, n_{0} = 67, 077;

nvalue[5] computes for t = 5 a random value, n_{0} = 7,885.

nvalue[6] computes for t = 6 a random value, n_{0} = 82,485.

nvalue[4] computes for t = 9 a random value, n_{0} = 69,965.

nvalue[1] computes for t = 1 a random value, n_{0} = 87, 381.

Remark: [tex]n_{0} \equiv 1[/tex] mod 4 easily generate infinitely many more values for each computed [tex]n_{0}[/tex] for a given index, t.

Dave.

[Guest]

Update: ... starting positive odd integers, [tex]21 \le n_{0} < 100,000[/tex] (approximately) ...

Reference Link: https://theory-of-energy.org/2020/09/17/a-brief-analysis-of-the-collatz-conjecture/.

Dave.

[Guest]

Update:

FYI: Mathematica code for finding some starting values (positive odd integers), [tex]21 \le n_{0} < 100,000[/tex] (approximately), for some indices,
[tex]1 \le t \le 10[/tex].

1. Wolfram Cloud (Mathematica Online), https://www.wolframcloud.com/; (Please login/register for a new/free basic account...)

2. Published Source Code Link, https://www.wolframcloud.com/obj/openmind123omega/Published/collatzcalc.nb.

Remark: The function, nvalue[t], computes a random positive odd integer, [tex]21 \le n_{0} < 100,000[/tex] (approximately), for a small index, [tex]1 \le t \le 10[/tex] (number of trials for a Collatz sequence of odd integers to converge to one from the computed [tex]n_{0}[/tex]).

Source Code:

nvalue[t_] := (
tt= t;
icnt = 0;
n = RandomInteger[{21, 100000}];
If[EvenQ[n], n = n + 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, [tex]n_{0} = 54, 613[/tex];

nvalue[10] computes for t = 10 a random value, [tex]n_{0} = 67, 077[/tex];

nvalue[5] computes for t = 5 a random value, [tex]n_{0} = 7,885[/tex].

nvalue[6] computes for t = 6 a random value, [tex]n_{0} = 82,485[/tex].

nvalue[4] computes for t = 9 a random value, [tex]n_{0} = 69,965[/tex].

nvalue[1] computes for t = 1 a random value, [tex]n_{0} = 87, 381[/tex].

Remark: [tex]n_{0} \equiv 1[/tex] mod 4 easily generates infinitely many more values for each computed [tex]n_{0}[/tex] for a given index, t.

Dave.

[Guest]

We update "[tex]21 \le n_{0} < 100,000[/tex] (approximately)" to "[tex]21 \le n_{0} < 100,000[/tex] (initially)" because the computed [tex]n_{0}[/tex] may exceed the bound, 100,000.

Remark: We can greatly increase that bound (100,000) for larger indices (t-values) but at the cost of much higher computation times.


Dave.

[Guest]

We update "[tex]21 \le n_{0} < 100,000[/tex] (approximately)" to "[tex]21 \le n_{0} < 100,000[/tex] (initially)" because the computed [tex]n_{0}[/tex] may exceed the bound, 100,000.

Remark: We can greatly increase that bound (100,000) for larger indices (t-values) but at the cost of much higher computation times.


Dave.

Hmm. We can develop a more efficient algorithm for computing "[tex]21 \le n_{0} < 100,000[/tex] (initially)" for given indices, t > 10.

Example: For t = 41, we have [tex]n_{0} = 27[/tex].

[Guest]

... Better yet, we can use a supercomputer and our dreams will come true! Go Blue!

[Guest]

FYI: nvalue[t] = {t, [tex]n_{0 }[/tex]}

nvalue[1] = {1, 21845}

nvalue[2] = {2, 233013}

nvalue[3] = {3, 77589}

nvalue[4] = {4, 6445}

nvalue[5] = {5, 68693}

nvalue[6] = {6, 40021}

nvalue[7] = {7, 57139}

nvalue[8] = {8,37685}

nvalue[9] = {9, 23693}

nvalue[10] = {10, 60225}

nvalue[41] = {41, 58131}

nvalue[22] = {22, 57457}

nvalue[17] = {17, 30461}

[Guest]

FYI: nvalue[t] = {t, [tex]n_{0 }[/tex]}

nvalue[1] = {1, 21845}

nvalue[2] = {2, 233013}

nvalue[3] = {3, 77589}

nvalue[4] = {4, 6445}

nvalue[5] = {5, 68693}

nvalue[6] = {6, 40021}

nvalue[7] = {7, 57139}

nvalue[8] = {8,37685}

nvalue[9] = {9, 23693}

nvalue[10] = {10, 60225}

nvalue[41] = {41, 58131}

nvalue[22] = {22, 57457}

nvalue[17] = {17, 30461}

Please see the attached file for the details of our source program (pdf file) for the above data... Enjoy! Dave.