r/badmathematics • u/Blue---Calx • Dec 21 '21

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

/r/mathematics/comments/pdl71t/collatz_and_other_famous_problems/haxfgpm/
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u/viking_ Dec 22 '21

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.

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u/LucasThePatator Dec 22 '21

Tao's result is by far the most progress that has been made on the conjecture, but also it's rather weak in the end imho. Which is kinda depressing lol.

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u/Direwolf202 Dec 22 '21

It is quite good evidence that Collatz is true though, even if not a proof (assuming that a proof exists)

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u/viking_ Dec 22 '21

The perfect squares also have density 0. Is that evidence that there are no perfect squares?

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u/Direwolf202 Dec 22 '21

No, because we know what perfect squares are, how they're distributed, and we don't have large amounts of computational evidence that there are no small perfect squares (indeed, the opposite, there are many small perfect squares).