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:
- mathoverflow.net/questions/75698/examples-of-seemingly-elementary-problems-that-are-hard-to-solve
- www.reddit.com/r/mathematics/comments/klev7b/whats_your_favorite_easy_to_state_and_understand/
- mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts this one is for proofs for which simpler proofs exist
- 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
A beautiful quote fom Edward Titchmarsh:[ref]Ciro Santilli believes that it is perhaps for this reason that simple to state but hard to prove theorems are so attractive.
It can be of no practical use to know that pi is irrational, but if we can know, it surely would be intolerable not to know.
This quote was brought to Ciro Santilli's attention at: www.quantamagazine.org/recounting-the-history-of-maths-transcendental-numbers-20230627/
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