After all efforts spent on wrong proofs of famous number theory conjectures and theorems like Goldbach's or Fermat's last theorem, could one find some simple statements (might be correct ones) whose proofs seemed trivial at the time but now are not trivial at all? Or that seemed true but turned out to be incorrect.
I'm thinking of something similar to "proofs" of the isoperimetric property of the circle or of Euclid's 5th postulate, which have been shown to contain logically equivalent assumptions to the postulate itself. What I also have in mind is: if arithmetic is incomplete can one find some simple intuitively true but unprovable facts about natural numbers?
Unfortunately, the only known true but unprovable facts in arithmetic are far from intuitive, or rather they are more about logic than about natural numbers, see How is a Godel sentence constructed?. But there are facts with short intuitive "proofs", whose actual proofs are much longer, Euler's are a typical example.
There are also many interesting properties of primes that follow from the intuitive idea that they are 'randomly distributed' known as Cramér's probabilistic model. Some were proved some not, but I doubt they are unprovable.