Edit: Based on the useful comments below. I edited the original post in order to seek for other important historical gaps in mathematics.
Mathematics is full of the historical gaps.
The first type of these gaps belongs to the statements which has been proved long after that mathematicians conjectured that they should be true.
The second type is equivalent statements and equivalence theorems (if and only if theorems). Sometimes detecting existence of an "equivalence" relation between two mathematical statements is not easy. In some cases a direction of a particular equivalence proof (if direction) is as immediate as a simple (or even trivial) observation but it takes many other years for the greatest mathematicians of the world to prove the other side (only if direction).
Here I'm looking for examples of these types of long historical gapes between statements and proofs in order to find a possibly eye-opening insight about the type of those critical mysterious points which make the gape between seeing the truth of a mathematical statement and seeing its proof such large.
Question 1: What are examples of conjectures which have been proved long after that they have been stated?
Question 2: What are examples of equivalence theorems with a large historical gap between their "if direction" and "only if direction" proofs?
Remark: Please don't forget to add references, names and dates.