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It seems that there can be many statements in mathematics that are easy to understand but very hard to solve, such as Fermat's Last Theorem, or the Twin Prime Conjecture. However these seem to be problems that are seen as important in the field of mathematics and were asked(or claimed to be solved) by distinguished mathematicians

I wonder then how easy would it be for anyone to ask an almost impossible to solve math questions(that is not already related to an existing conjecture). Like how many "easy" to understand questions on StackExchange or ones asked in research are really hard to solve(by hard to solve, I mean at-least on the level of difficulty as Fermat's Last Theorem) and not just something that just requires a lot of research and problem solving by a single person.

Also is their research on how many such "easy" to understand questions are asked regularly.

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It's very easy to ask very hard questions. For example, take some property $P$ of natural numbers and ask "are there infinitely many natural numbers with property $P$?". Depending on the property, it may be easy to show that there are infinitely many, or it may be easy to show that there are only finitely many. But if neither of these is very easy, it's quite likely to be an extremely hard question, perhaps even one that's undecidable (although in that case you are also unlikely to be able to prove that it is undecidable). If it turns out to be possible to show there are infinitely many, you might add some new requirement to make it harder.

An example, chosen more or less randomly from the OEIS:

  • Numbers $k$ such that $3^k - 2^k$ is prime.
  • Numbers $k$ such that the sum of the squares of the first $k$ primes is divisible by $k$.
  • Numbers $k$ such that $ x^k + x + 1$ is irreducible over $GF(2)$.
Robert Israel
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