The Largest Gaps in the History of Mathematics

Question

Edit: Based on the useful comments below. I edited the original post in order to seek for other important historical gaps in mathematics.

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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.

Answer 1

$a^n+b^n=c^n$ has non-trivial integer solutions if and only if $n\le2$

Answer 2

A closed 3-manifold is homeomorphic to a 3-sphere if and only if it is simply connected. (Here the only if part is easy, and the if part is the relatively recently solved Poincare conjecture).

Answer 3

A non-trivial group of odd order is simple if and only if it is of prime order.

The part that a group of prime order is simple is "easy" since by Lagrange's Theorem it has no proper subgroups. The first complete proof of this was given in 1801.

The part where a simple group of odd order has prime order comes from the fact there are no non-abelian finite simple groups of odd order. This was proved in 1962-1963 by Walter Feit and John Griggs Thompson.

Answer 4

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.

Answer 5

Conjectured by Euler and Legendre and proven by Gauss:

$$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$

Answer 6

The Kepler conjecture:

No arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.

The proof was overwhelmingly complex. Only recently a formal proof has been finished.