Why are prime numbers so notoriously hard to write and find proofs for?

Take the twin prime conjecture:

There are an infinite amount of consecutive primes.

ie 11&13 ; 41&43 ; 197&199

This is an open question in math and hasn't been proved or disproved yet. There are however A LOT of open questions about prime numbers.

I would like to see some solved and or proven conjectures using primes to see how easy / hard it is to write such proofs.

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