Is the Collatz Conjecture solvable?

[Guest]

Reference link:https://en.wikipedia.org/wiki/Collatz_conjecture.

Yes! The Collatz Conjecture is solvable and true. Here's why:

Probability(Collatz sequence does not converge to one) = [tex]\prod_{m=1}^{\infty }\sum_{i=1}^{l_{m}}(1/2)^i\rightarrow 0[/tex].

Therefore, Probability(Collatz sequence does not converge to one) is zero.

Notes:

Probability[tex]( 4\mid n_{m } : l_{m }) = \sum_{i=1}^{l_{m}}(1/2)^i = 1 - 2^{-l_{m}}[/tex];

[tex]n_{m } = 3*n_{m-1 }/2^{l_{m-1 }}+1[/tex];

[tex]n_{m-1 }/2^{l_{m-1 }} \in 2\mathbb{N} - 1[/tex];

[tex]n_{m }/2^{l_{m }} \in 2\mathbb{N} - 1[/tex];

[tex]n_{1 } = 3*n_{0 }+1[/tex] for any [tex]n_{0 }\in 2\mathbb{N} + 1[/tex];

[tex]m, l_{m } \in \mathbb{N}[/tex] and [tex]l_{0 } = 0[/tex].

[Guest]

https://www.quora.com/Is-the-Collatz-Conjecture-solvable/answer/David-Cole-146

[Guest]

Mathematica 11 software function f[] for simulation of data:

f[m_] := (

n = m;

Print[n];

prob = 1.;

While[n > 1, n = 3*n + 1;

x = n; i = 0;

While[

EvenQ[x], ++i;

x = x/2];

s = 1 - 2^(-i);

Print[x];

prob = prob * s;

n = x;

];

Return[{m, prod}]

);

Example:

f[n_0 = 1367] yielded a probability of 0.0245548 before the Collatz sequence of odd integers,

{1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 1},

converged to one.

*****

[Guest]

Mathematica 11 software function f[] for simulation of data:

Mathematica 11 software function f[] for simulation of data:

f[m_] := (

n = m;

Print[n];

prob = 1.;

While[n > 1, n = 3*n + 1;

x = n; i = 0;cnt = 0; icnt = 0;

While[

EvenQ[x], ++i;

x = x/2];

s = 1 - 2^(-i);

cnt = cnt + 1;

If[i == 1, icnt = icnt + 1];

Print[x];

prob = prob * s;

n = x;

];

Return[{m, prob, cnt, icnt, 100. * icnt/cnt}]

);

Example 1:

f[n_0 = 1367] yielded a probability (prob) equal to 0.0245548 before the Collatz sequence of odd integers,

{1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 1},

converged to one;

Example 2:

{n_0 =1917424378915227003508542412844973284190406292736261788098013,

prob = 2.14803*10^-104,

cnt = number of generated odd integers of Collatz Sequence =541,

icnt =number of times l_m = i equals one = 269,

icnt/cnt = 49.7227%};

Example 3:

{n_0 =1134322358892035783221922112828664036984286481192427929865901,

prob =1.30642*10^-113,

cnt = number of generated odd integers of Collatz Sequence =573,

icnt =number of times l_m = i equals one = 301,

icnt/cnt = 52.5305%}.

Examples above are empirical affirmation of the theory proving the Collatz Conjecture.

*****

[Guest]

An Important Note:

The key ideas of the above proof of the Collatz Conjecture is that [tex]4\mid n_{m }[/tex] dominates (> 50%) the Collatz processing when [tex]l_{m } > 1[/tex], and it is the same (50%) when [tex]l_{m } = 1[/tex]. Together the ideas means the Collatz sequence will always converge to one.

[Guest]

"The Collatz Conjecture is a wonderful play (contest) mainly between the operations of division and multiplication over the natural numbers. And the ideas of sequence and convergence are included in the plot. And the outcome is always a convergence of the Collatz sequence to one. Basically, it is the division by four or by [tex]2^{k + 2}[/tex] with k > 0 that overcomes the multiplication by three in the Collatz process that decides the outcome and the truth of the conjecture." -- David Cole

[Guest]

An Important Note:


Hmm. The value [tex]l_{m }[/tex] =1 will occur as often as [tex]l_{m }[/tex] > 1 according to the Law of Large Numbers. That fact explains any growth in the Collatz sequence, and it also explains why the Collatz sequence will always converge to one. -- Dave

[Guest]

An Important Note:


Hmm. The value [tex]l_{m }[/tex] =1 will occur as often as [tex]l_{m }[/tex] > 1 according to the Law of Large Numbers. That fact explains any growth in the Collatz sequence, and it also explains why the Collatz sequence will always converge to one. -- Dave

Update:

An Important Note:

The value, l_m =1, which implies division by two in the Collatz process will occur as often as all other values, l_m > 1, which imply division either by four or by eight or by sixteen or by thirty-two, … in the Collatz process according to the our probability calculations and according to the Law of Large Numbers. Thus, we expect the division by two will occur 50% of all possible divisions in the Collatz processing. That fact explains any growth in the Collatz sequence, and it also explains why the Collatz sequence will always converge to one.

[Guest]

An Important Note:


Hmm. The value [tex]l_{m }[/tex] = 1 will occur as often as [tex]l_{m }[/tex] > 1 according to the Law of Large Numbers. That fact explains any growth in the Collatz sequence, and it also explains why the Collatz sequence will always converge to one. -- Dave

An Important Note Update:

The value, [tex]l_{m} = 1[/tex], which implies division by two in the Collatz process will occur as often as all other values combined, i.e., [tex]l_{m} > 1[/tex], which imply division either by four or by eight or by sixteen or by thirty-two, … in the Collatz process according to the our probability calculations and according to the Law of Large Numbers. Thus, we expect the division by two will occur 50% of all possible divisions in the Collatz processing. That fact explains any growth in the Collatz sequence, and it also explains why the Collatz sequence will always converge to one.

[Guest]

Relevant Reference Link:

'Solution to Collatz's Conjecture' by Prof. J. W. Porras,

https://www.researchgate.net/publication/325389892_Solution_to_Collatz's_Conjecture.

[Guest]

FYI: 'Mathematician Proves Huge Result on ‘Dangerous’ Problem
By
KEVIN HARTNETT
December 11, 2019

Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades.'

https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/.

Hah! Any 'almost proof' of Collatz Conjecture by T. Tao (a so-called math expert on Collatz Conjecture, etc.) et al. is not a true proof! And any 'almost proofs' of the Riemann Hypothesis or the Goldbach Conjecture are not true proofs!

The Collatz Conjecture, Riemann Hypothesis, and the Goldbach Conjecture are true!!!

-- David Cole,

https://wwww.researchgate.net/profile/David_Cole29/amp.

"Don't pay attention to "authorities" (so-called experts)!
Think for yourself!" -- Richard Feynman.

[Guest]

FYI: 'Mathematician Proves Huge Result on ‘Dangerous’ Problem
By
KEVIN HARTNETT
December 11, 2019

Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades.'

https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/.

Hah! Any 'almost proof' of Collatz Conjecture by T. Tao (a so-called math expert on Collatz Conjecture, etc.) et al. is not a true proof! And any 'almost proofs' of the Riemann Hypothesis or the Goldbach Conjecture are not true proofs!

The Collatz Conjecture, Riemann Hypothesis, and the Goldbach Conjecture are true!!!

-- David Cole,

https://wwww.researchgate.net/profile/David_Cole29/amp.

"Don't pay attention to "authorities" (so-called experts)!
Think for yourself!" -- Richard Feynman.

Moreover, the difficulty of proving/understanding the Collatz Conjecture has been exaggerated by Erdos, Lagarias, et al. That indicates their lack of understanding the conjecture/problem. And therefore, they are so-called experts on the Collatz Conjecture. How can anyone solve a problem if one does not understand the problem?

Remarks: An 'almost proof' here indicates an incomplete/flawed proof. Math proofs of important math conjectures should always be complete/true...

Dave.

[mugbuff]

n and 4n+1 converge to the same value after one 3n+1 step. thus no need to test any 4n+1 integer, i.e. 1/4 of all integers
2n and 16n+3 converge after 2 steps )
4n+3 and 32n+28 converge after 2 steps ) No need to test another 1/16 and 1/32 of all integers
I can produce similar facts up to convergence after 7 steps which indicate that the number of integers need testing decreases with each step after this by a factor of 13/16, i.e. tending to zero. To provide similar facts for more steps is easy except that the number of such formats increases by a factor of 3 for each step up and is thus extremely laborious.

[uthankran]

Hello!

I just went through two different Wikipedia articles on the Collatz conjecture and I'm still confused regarding its status as "unsolved". I feel like maybe I'm misunderstanding the mathematical usage of the term "solved", but the conjecture seems really straight-forward and easy to test (they even had a really simple pseudocode example of how you'd test it!). So why hasn't it been tested to a satisfactory degree?

My only guess is that the idea is "well, there may be another (much larger?) number somewhere that doesn't fit the conjecture", in which case I wonder, can the conjecture be proven? Because there's always going to be another number that hasn't been tested. At what point could we consider it to be proven / solved?

[Guest]

Hello!

I just went through two different Wikipedia articles on the Collatz conjecture and I'm still confused regarding its status as "unsolved". I feel like maybe I'm misunderstanding the mathematical usage of the term "solved", but the conjecture seems really straight-forward and easy to test (they even had a really simple pseudocode example of how you'd test it!). So why hasn't it been tested to a satisfactory degree?

My only guess is that the idea is "well, there may be another (much larger?) number somewhere that doesn't fit the conjecture", in which case I wonder, can the conjecture be proven? Because there's always going to be another number that hasn't been tested. At what point could we consider it to be proven / solved?

Please learn to think yourself and research/learn how to solve problems...

[Guest]

... think for yourself ...

[Guest]

https://www.researchgate.net/publication/351347153_COLLATZ_CONJECTURE_-THE_PROOF