Proof of the Collatz hypothesis

3r3-31. I will continue the started. previous 3r331. post topic proof

Collatz's conjecture .

What is this hypothesis? Take any natural number

. If it is even, then divide it by

and if it's odd, then multiply by

and add

3r3146.

(we receive

). On the resulting number, perform the same actions, and so on. Collatz's conjecture is that whatever the initial number is 3r-3254.

we neither took, sooner or later we will get a unit.

3r330. 3r3331.

#### Proof of the Collatz hypothesis

We reformulate the hypothesis as follows: take any positive integer

. If it is even, then divide it by

until it loses the parity property, and then we transfer it to the number system on the base

and add

3r3146.

until the base of the number system becomes inverse

, at each step multiplying

at

where

3r362.

- number of the current step. Collatz's conjecture in such a formulation is that whatever the initial number is 3r-3254.

we did not take it, sooner or later it will happen, that is, the equation is 3r3-33132.

3r33132.

where

- The number of odd steps,

- the total number of steps, has a solution for any natural 3r33254.

which is obvious so.

**Proven**.

But nothing is clear, especially why I reformulated the hypothesis in this way and where did the number of steps come from in it. Maybe I'm wrong? Let's check on specific examples from the Wikipedia page.

For the number

we get

3r33132.

3r33132.

The solution is 3r33254. 3r33112.

, that is, the number

can convert to

3r3146.

in three odd steps, and all it takes is

steps. Right.

For the number

3r3143.

we get

3r33132.

3r33132.

3r3133.

The solution is 3r33254. 3r33140.

, that is, the number

3r3143.

can convert to

3r3146.

in seven odd steps, and all you need is

3r3149.

steps. That's right too, but still nothing is clear. The answer is below.

Attention! Nervous and spoiled Russian mathematical education, please do not read further!

#### Some basic concepts of the theory of entropy 3r3r2202.

System is an indefinable concept, on the basis of which unprovable definitions are built: 3r-32r.

Entropy - information capacity of the system.

Extropy is the opposite of entropy.

Phase transition - a decrease in entropy by one order of magnitude, which occurs with the accumulation of sufficient and necessary amount of knowledge.

The evolution of a small n

Consider a specific example: take from the formulation of the hypothesis n and turn it into 3r-3254.

. How to achieve this? Reducing ignorance, that is, asking questions that are possible only a definite answer "yes" or "no." Let's go:

1. n - the plane? No. This answer reduced our ignorance about n to the knowledge of “n is not a plane”, but did not tell us anything about the properties of n, that is, it reduced the entropy, but did not increase extropy.

2. n - a mathematical object? Yes. This answer has increased the extropy, now we know that n is a mathematical object, therefore it has all the properties of a mathematical object, in particular, it is a variable or a constant.

3. n - constant? No. This answer again reduced the entropy and made a phase transition. The amount of accumulated information "n is a mathematical object" and "n is not a constant" turned into its quality - output "3r-3254. 3r-?215. 3r-?256. - variable" and now allows us to reduce the entropy of the data above definitions.

Entropy (in mathematics) is an inherent property of a mathematical object, a measure of our ignorance about it as a system, a value measured in bits of entropy. Informally: the answer is "no" to the question.

Extropy (in mathematics) is the opposite of entropy (in mathematics), a measure of our knowledge of a mathematical object as a system, a quantity measured in extropy bits. Informally: the answer is "yes" to the question.

Phase transition (in mathematics) - a decrease in entropy (in mathematics) by one bit of entropy, which occurs with the accumulation of a sufficient and necessary amount of knowledge.

A little magic theory of entropy 3r3r2202.

So how did I get my equivalent of the Collatz conjecture? Suppose that initially

It had the appearance

that is, contained

bits of extropy: “yes” answers to questions “is 3r32544 divided.

at

respectively? ”and a certain number of bits of entropy, defined by the variable

. The division operation by

we lowered extropy three times, in the end it became equal to zero, the entropy remained unchanged and equal to 3r-3254.

where

- the number of digits in the binary notation of the number

. By transferring to a number system with a fractional basis, we each time made a phase transition (in mathematics), because we got knowledge of "3r35454. 3r-3636. 3r-?356. Is divided into 3r3 -3254. 3r-33239. 3r-?256." left and replaced zero in the rightmost bit by one. As a result,

shifts from a completely undefined number, we got the numbers from 3r33254.

digits, all the numbers in which the zeros, that is, the number is completely defined. Adding to the zero units we got the desired.

P.S. The thoughtful reader must have noticed that the most important consequence of the proof of the hypothesis is the fact that 3r-3254.

the number is always compound for any

$ 2 "data-tex =" inline "> 3r33256. This is the most important discovery, the key to the exponential reduction of the computational complexity of the computer problem of factoring large numbers. I will definitely develop this topic in the next post, which, apparently, because of my negative karma, will see the light only in a week.

3r33267.

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