The seventh most popular easily understood unsolved problem on MathOverflow

> The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^361.[3]

> An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980.[5]

EDIT: rephrased because despite having a maths degree I can't count.

[0]: https://www.youtube.com/watch?v=NOCsdhzo6Jg

[1] https://en.wikipedia.org/wiki/Collatz_conjecture?useskin=vec...

why on earth would you ever think that

The prime number theorem states that the prime counting function $\pi(x)$ is well approximated by the integral $\int_0^x \frac{dt} {\log t},$ which is a function that's become named $\mathrm{li}(x).$ Littlewood proved that the sign of $\pi(x) - \li(x)$ changes infinitely often, but his proof didn't produce a specific value where such a sign change occurs.

In 1933, one of Littlewood's students showed that, assuming the Riemann hypothesis, at least one sign change occurs below the number $e^e^e^79$, which is approximately $10^10^10^34.$ In 1955, he was able to show unconditionally that such a sign change occurs below the number $e^e^e^e^7.705$, which is approximately $10^10^10^964$.

Both of those numbers absolutely dwarf the estimated number of elementary particles in the observable universe, so it's utterly impossible to either compute or write them down, even if you used the entire universe as your computer or your writing surface.

https://en.wikipedia.org/wiki/Skewes%27s_number

That upper bound has since been lowered in 1999 to 1.38922 * 10^316 here: https://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-...

I imagine it there might have been a bit more progress in the last 25 years, but I have no idea if it's possible to actually write down all the digits of the current best upper bound.

> the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980.[5]

I wonder what the most pleasing equivalent formulation of the RH would be. Gotta assume that some of them are much more directly about prime numbers. Yours seems pretty good. There are some others on here: https://mathoverflow.net/questions/39944/collection-of-equiv... which seem good also.

The million dollar prize rules are surprisingly strict. They expect a "natural" proof, in that writing a one-page paper "my computer found counterexample 2^1729-1" may not get you the full prize, maybe instead a tiny acknowledgement prize, and they will wait for a proof of a generalization of the problem. https://www.claymath.org/wp-content/uploads/2022/03/millenni...

> level 2: if not provable (can't prove level 0) then can't prove undecideable. (Goldbach, Collatz)

If you're operating in the ordinary naturals, neither Goldbach nor Collatz are undecidable. The statements are either true or false. The question at hand is whether we can generate a finite proof of that fact in a given axiomatic system. The quote at hand simply said that if no such proof exists (obviously implying the statements are true because otherwise a simple finite proof would be a counter-example) then similarly no finite proof of that lack of existence exists either.

They might be undecidable in other arithmetic systems (implying that some models of those systems would have the statements be true and some would have them be false), but not for the naturals.

> Which means there's no level infinity, since there's no level infinity minus 1 to talk about so it's meaningless. Correct?

When you're making up a new definition, you care about (1) is it coherent (don't want to be like the proverbial Ph.D. who made up an exciting mathematical object and studied it for years before finding out no such objects could exist), (2) what can you deduce about that object, and (3) is it "useful" (for some broad definition thereof). The question of "meaningfulness" can lead you down incorrect paths because it merges (2) and (3). The definition you picked for the broad concept you have in your mind has no level "infinity" because you snuck the naturals into the definition as the indexing set (defined using the successor function). If we strictly look at point (2), yes, there's no "infinity".

The general concept you're looking at though might more naturally fit in a world where there are infinities (i.e., let's look at point (3) a bit more). As a rule of thumb, if you're looking at a set of things indexed by the naturals, especially if they have a successor function, especially if they're defined inductively, especially if their interesting properties are defined in terms of all previous items, it's natural to take a look at indexing via the ordinals and trying to define them via transfinite induction.

For this particular set of things, that may or may not be straightforward. The notions of unprovability, undecidability, ... are a bit intricate, and you need to define them with respect to the system being studied and the system being used to study them. Right now, your inductive definition _seems_ to also require a predecessor function (which, if intrinsic to the property being analyzed would preclude many attempts to wrangle in some infinities), but I wouldn't be surprised if a careful re-writing found that to be an extraneous detail, in which case you could add in a limiting case (defining f(w) in terms of f(n) for all n<w for all limit ordinals w).

But in this case, I'd bet that developing the math needed to solve it would end up with practical consequences. I think of your question sort of like "what are the practical applications of teaching an AI to master Starcraft?" Even if playing Starcraft isn't practical, the existence of such an AI implies lots of practical consequences.

…or maybe the fact that it proves so hard to answer shows that we’re just wrong for thinking that it’s ‘elementary’. Therefore, any solution would necessarily require new breakthroughs that would lead to greater general understanding. So:

1) mathematicians would like to save face

2) you can’t solve difficult problems without accidentally solving other difficult ones that turn out to have immediate ‘practical’ importance

And a telephone company wanted to connect calls more easily, so they invented the transistor, which changed the world in unimaginably many ways. Sometimes you just don’t know where you’ll end up until you try it.

I can’t tell if you’re just joking. Proofs of facts that seem obvious or irrelevant are very important not for the yes-no question they answer but for the development of the deep mathematical theory they necessitate.

Did you miss the infinite sum there? How would you prove an infinite sum equals a transcendental number by counting finite bits? You'd have to count infinite bits.