{tag=Simple to state but hard to prove}
{wiki}

It is cool how even for such a "simple looking" problem, we were still unable to prove optimality as of 2020!


3SUM

{c}
{tag=Simple to state but hard to prove}
{title2=1937-}
{wiki}

Given stuff like https://arxiv.org/pdf/2107.12475.pdf on <Erdős' conjecture on powers of 2>, it feels like this one will be somewhere close to <computer science>/<Halting problem> issues than <number theory>. Who knows. This is suggested e.g. at <The Busy Beaver Competition: a historical survey by Pascal Michel>.


Collatz conjecture

{tag=Simple to state but hard to prove}

https://en.wikipedia.org/wiki/Transcendental_number#Conjectured_transcendental_numbers

There's a billion simple looking expressions which are not known to be transcendental numbers or not. It's cute <simple to state but hard to prove> at its best.

Open as of 2020:
* $e + \pi$

Bibliography:
* https://www.quantamagazine.org/recounting-the-history-of-maths-transcendental-numbers-20230627/ How Math Achieved Transcendence by David S. Richeson (2023).

\Video[https://www.youtube.com/watch?v=BdHFLfv-ThQ]
{title=Why π^π^π^π could be an integer by Stand-up Maths (2021)}
{description=Sponsored by Jane Street. <finance is a cancer of society>[Shame].}


Transcendental number conjecture

One of the most beautiful things in mathematics are theorems of conjectures that are very simple to state and understand (e.g. for <K-12>, lower <undergrad> levels), but extremely hard to prove.

This is in contrast to conjectures in certain areas where you'd have to study for a few months just to precisely understand all the definitions and the interest of the problem statement.

Bibliography:
* https://mathoverflow.net/questions/75698/examples-of-seemingly-elementary-problems-that-are-hard-to-solve
* https://www.reddit.com/r/mathematics/comments/klev7b/whats_your_favorite_easy_to_state_and_understand/
* https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts this one is for proofs for which simpler proofs exist
* https://math.stackexchange.com/questions/415365/it-looks-straightforward-but-actually-it-isnt this one is for "there is some reason it looks easy", whatever that means


Ciro Santilli @cirosantilli 37

 Tagged: Simple to state but hard to prove