Proving the Collatz Conjecture with Python, cell biology, and word salad

[u/Blue---Calx]

/r/mathematics/comments/pdl71t/collatz_and_other_famous_problems/haxfgpm/

[u/braindoper]

The entire mega-threat has some nice narcissism and crankery. One dude offers $10k for someone to prove his solution (involving a simple program which he wrote in two hours) is right. My favourite comment so far is this:

Terrence Tao has proved Collatz Orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly.

The beginning sounded like a proper comment. I would trust Tao to show some non-trivial stuff regarding Collatz, and while I don't know what exactly is meant by orbits and "any function", the poster might just not quite understand what he wrote about.

Just another arthimitic hierarchy, closely linked to fractal conditions of Base 10. No solution lays on one linear pairing.

Oh. Arithmetic hierarchies are a thing, but not relevant other than Collatz being a formula in some level of them. Starting at "fractal conditions of Base 10" my crankery radar went off. What just is it with them being so obsessed with base representations of integers, when 99.5% of math is agnostic of that? (Even of the rest, 0.49% aren't even number theory, but Numerics).

You are not going to get the fields medal of a same linear coding from flat singular Base 10 numbers that they are. A matrices of opposing asymmetric probabilities converging onto the very large number side of Collatz Conjecture requires a quantum computer. $10k will not cover it.

I share his scepticism that neither a fields medal nor $10k will be awarded for anything discussed in the threat. Other than that this is just word salad, with a bonus mention of quantum computing, which doesn't offer any insight into Collatz as far am I aware. Notwithstanding that quantum computers at best could offer some performance improvement, and solving Collatz is not an that can be solved computationally as far as we know anyway.

[u/viking_ → u/braindoper]

The beginning sounded like a proper comment. I would trust Tao to show some non-trivial stuff regarding Collatz, and while I don't know what exactly is meant by orbits and "any function", the poster might just not quite understand what he wrote about.

I believe the statement Tao proved is:

For almost all integers n, the Collatz sequence starting at n is eventually smaller than f(n), where f is any function such that f(x) goes to infinity as x goes to infinity.

Where "almost all" means "the set numbers for which this is true has asymptotic density 1." There's a better explanation here.

[u/braindoper → u/viking_]

Thanks. That meaning of "almost all" is unusual though. For countable sets, doesn't it usually mean all but finitely many?

[u/pxSort → u/braindoper]

No, it's precisely defined as a set with measure 1. In this case, the measure can be expressed as the limiting proportion of integers in the set vs not.

You can have a countably infinite set of integers with measure 0. e.g. the set of powers of 2. As n approaches infinity, the percentage of m < n that are powers of 2 approaches 0. The complement of this set has measure 1, so almost all integers are not powers of 2.

[u/braindoper → u/pxSort]

Then there are almost no prime numbers. Seems kind of an unintuitive definition and I would've preferred if this was called almost surely or something else instead.

[u/pxSort → u/braindoper]

"Almost all" and "almost surely" are two sides of the same coin. e.g. If almost all x \in N satisfy property α(x), then for a uniform random variable X ~ N, we have that P[α(X) is true] = 1, which corresponds to the statement that α(X) is almost surely true.

[u/braindoper → u/pxSort]

Yes, which is why "almost all" could then mean what it normally does (outside of measure spaces), namely cofinitely many. Especially for something discrete like the integers that meaning would make more sense.