How are there unsolved math problems?
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Most of the things that we consider unsolved math problems are _conjectures_. That basically means they’re educated guesses about patterns or relationships (between numbers) that may or may not exist. However, just because someone guesses that this relationship may exist doesn’t mean it’s proven to exist. And the guess itself may not provide strong guidance on how to prove it.
For example, I may say to you that I conjecture that for _every_ prime number you find, I can find a bigger one. This would be considered unsolved until someone either proved me right or wrong.
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