I found an article by Peter Schorer from June 29,2015 which is claming to give a solution of the 3x+1 problem. Are there remarks from any mathematicians if this is correct or not?
In addition to Avva’s anwser, let me point out what is the problem with the linked proof.
In the document various objects are defined and used, but the structure of the proof boils down to essentially the following. Consider the map $C$ from the positive odd integers to the positive odd integer that maps $x$ to $(3x+1)/2^a$, where $a$ is the exponent of 2 in $3x+1$, and let $J$ be the set of all odd positive integers that map to 1 under repeated application of this map $C$. We note that, for any odd integer $x$, the truth or falsehood of the statement $x\in J$ depends only on $x$, and is in particular independent of whether or not there exist counter-examples to the Collatz conjecture. Hence, the set $J$ is independent of whether or not there exist counter-examples to the Collatz conjecture. Clearly, if counter-examples do not exist, then $J$ is simply the set of all odd positive integers. But by the independence of $J$, it follows that $J$ is still the set of all odd positive integers if counter-examples do exist, contradicting the definition of $J$. Hence, the assumption of the existence of counter-examples leads to a contradiction, so no counter-examples to the Collatz conjecture exist, so the Collatz conjecture must be true.
Clearly this proof cannot be correct, as it makes essentially no reference to the actual Collatz function $C$ or the number 1. In other words, the same ‘proof’ would show that (for example) any number would go to 731 under repeated application of the map $C$, or for that matter under repeated application of the map $x\mapsto x^2$.
It is hard to pinpoint exactly what goes wrong in the proof, but it seems to be that the statement “$J$ does not depend on the truth of the Collatz conjecture” does not have mathematical content. It is similar to saying that the value of 6 does not depend on the value of 2, and then arguing that if $2=3$ we must have $6 = 2+2$, and therefore also when $2\neq 3$ we have $6=2+2$. It is clear that this reasoning does not hold.
So to answer your question, no, the linked pdf does not solve the Collatz conjecture.
It is not solved. Schorer’s been presenting various faulty proofs for more than 10 years, and whenever mathematicians checked them they were unimpressed.
This remark by a mathematician is from 2009 but there’s no reason to think things have improved since:
The claimed proof that the conjecture is unprovable, in an article by Craig Alan Feinstein, is also clearly bogus. It assumes without good reason that any proof must explicitly “specify the formula” for the $k$-th iteration of the function, and makes other elementary mistakes.
I see several red flags here.
I don’t know the author, but the paper has the marks of a crank written all over it.
An additional information on Craig Alan Feinstein’s article:
In an email that I sent to him on 26 Aug 2008, I wrote the following:
I came across your article, suggesting that the $3n+1$ conjecture is unprovable.
I have not been able to find a counterexample to (what I think is) your main argument in the article:
For any $k$ and $n$, every proof that $T_k(n)=1$, should be at least $k$ bits long.
Nevertheless, it seems to me that this argument takes in consideration only the constructive type of proofs (unlike proofs by negation, for example).
So for instance, we can assume the following:
There exists an $n$, for which $T_k(n)\neq1$ for all possible values of $k$.
And then show that it violates some other (proved) theorem.
In this type of proof, we may not need to build a solution (that is $k$ bits long) at all.
I would appreciate your comment on that.
I would also like to know if this article has been formally accepted as a proof that the $3n+1$ conjecture is unprovable (in order to know if there is any point trying to prove it).
Thank you very much for your time.
In his response on the same day, he wrote the following:
Thank you for your question.
Go through every step of the logic of my proof.
If you can’t find any holes in the logic, then you have to conclude that there are no other types of possible proofs of the Collatz conjecture.
The important thing to understand in all of this is that proofs do not work by magic.
You could suggest all sorts of hypothetical ways of proving Collatz but this doesn’t mean they will work.
As for formal acceptance of my proof, I don’t think there is such a thing.
My proof is either right or wrong regardless of who believes it or not.
Google a paper I wrote, “Complexity Science for Simpletons”, published in a physics journal, which also talks about the Collatz conjecture and also the Riemann Hypothesis.
I hope that it shed some light on the validity or invalidity of this article.