## Does the paper "Proof of Collatz's Conjecture" by Agelos Kratimenos contain valid proof?

The document "Proof of Collatz's conjecture" (link arXiv) by Agelos Kratimenos was submitted to arXiv a few days ago (4 Nov. 2019).

Is it really a valid proof?

I mean, I know that arXiv is a prestigious newspaper. Do they have peer reviews before an article is accepted? Or are submissions possible without that?

It would be amazing if it was really a proof of Collatz's conjecture.

## number theory – solving two variables for n related to Collatz's conjecture

For this code, for each x, I want to solve all ranges of values ​​for c1 and c2 in a limited range, ie. C1 and c2 in the range of real numbers + -100 for c1 and c2 for each x, which gives "Length (stepsForEachN) == nRangeToCheck – 1". Here is the code up to now, I'm not sure how to solve both variables c1 and c2 for each x:

``````(*stepsForEachN output is A006577={1,7,2,5,8,16,3,19} if c1=c2=1*)
c1 = 1;
c2 = 1;
nRangeToCheck = 10;
stepsForEachNwithIndex = {};
stepsForEachN = {};
stepsForEachNIndex = {};
maxStepsToCheck = 10000;

c1ValuesForEachN = {};

For(x = 2, x <= nRangeToCheck, x++,

n = x;

For(i = 1, i <= maxStepsToCheck, i++,
If(EvenQ(n), n = Floor((n/2)*c1),
If(OddQ(n), n = Floor((3*n + 1)*c2))
);

If(n < 1.9,
AppendTo(stepsForEachN, i);
AppendTo(stepsForEachNIndex, x);
AppendTo(stepsForEachNwithIndex, {x, i});
i = maxStepsToCheck + 1
)
)
)
Length(stepsForEachN) == nRangeToCheck - 1
``````

## Fourier Analysis – Explicit Limits of Tao's Result on Collatz's Conjecture

A new pre-print of Terry Tao has been published recently and has yielded interesting results on the topic of Collatz's conjecture. I will not cite the precise result, but rather an equivalent formulation that Tao notes in his remark 1.4:

For all $$delta> 0$$ there is a constant $$C_ delta$$ such as $$mathrm {Col_ {min}} (N) leq C_ delta$$ for everyone $$N$$ in a subset of $$mathbb N + 1$$ at least logarithmic density $$1- delta$$.

My question is something about the growth rate of $$C_ delta$$ as $$delta to 0$$. I will ask two specific questions in this regard. The first, I imagine, could have been known before the recent Tao result.

Are there values ​​of $$delta <1$$ for which a explicit upper limit for $$C_ delta$$ is known?

The second essentially asks if something explicit can be deduced from the result of Tao.

Is the function $$N mapsto C_ {1 / N}$$ Upper bound by a computable function?

(Note that I do not think the answer is "obviously yes" to the mere existence of $$C_ delta$$since the control of the operation of a particular value seems to be $$Sigma ^ 0_3$$ (it exists $$N_0$$ as for all $$N> N_0$$ it exists $$M$$ as in $$M$$ not at least $$1- delta$$ numbers below $$N$$ to go downstairs $$C_ delta$$) and is obviously not lower)

## A new way to look at Collatz's conjecture

Firstly, it is not a proof, but a new way of looking at Collatz (which other mathematicians, hopefully, will be able to prove / counter-test). I came here to find out if what I found is valid.