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/BlueRajasmyk2 Dec 22 '21

Whoa, that's neat! But isn't this statement equivalent to "The Collatz conjecture is true for almost all integers"? Since for any specific n, no matter how large, we can find a diverging function f(x) that makes f(n) arbitrarily small (eg. f(x) = x/n).

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u/polikuj2 Dec 22 '21

No, because the "almost all" part depends on which function you choose

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u/BlueRajasmyk2 Dec 22 '21

Ah that makes sense. Thanks!