The Largest Gaps in the History of Mathematics 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.


Finding explicit formulas for the roots of polynomials:
-The linear and quadratic cases were known to the Babylonians. The Greeks reportedly tried to solve the cubic without success.
-The solution of cubic and quartic equations were major discoveries in the Renaissance 
-The the lack of general formulas when the degree is larger than 4 was proved by Abel in the 19th century
The gaps and subsequent solutions saw a shift from geometric methods (the original quadratic formula was actually a ruler/compass construction) towards purely algebraic ones. This also led to the initial development of complex numbers and abstract group/field theory.
