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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.

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    $\begingroup$ Community wiki? $\endgroup$
    – Martin Brandenburg
    Commented Nov 2, 2014 at 22:03
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    $\begingroup$ There is nothing mysterious about this. Consider for example "A problem is in P if and only if it is in NP". One way is trivial; the other way is the most important open problem in computer science. $\endgroup$
    – MJD
    Commented Nov 2, 2014 at 22:03
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    $\begingroup$ This question is not really different from "what is the longest time between a conjecture and its proof". $\endgroup$
    – vadim123
    Commented Nov 2, 2014 at 22:05
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    $\begingroup$ Maybe this question is better suited at History of Science and Mathematics? $\endgroup$
    – Ali Caglayan
    Commented Nov 2, 2014 at 22:06
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    $\begingroup$ There are also quite a lot of (conjectured) equivalences where one direction is still unknown (Kaplansky's conjectures, P vs NP, Jacobian conjecture). $\endgroup$
    – Martin Brandenburg
    Commented Nov 2, 2014 at 22:25

6 Answers 6

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

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    $\begingroup$ This answer approves vadim123's comment "This question is not really different from "what is the longest time between a conjecture and its proof"". I think more interesting answers should be equivalences $A \leftrightarrow B$ where $A,B$ are really "similar" in some sense (for example, the Poincaré conjecture). $\endgroup$
    – Martin Brandenburg
    Commented Nov 2, 2014 at 22:09
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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).

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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.

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  • $\begingroup$ A priori the Feit-Thompson Theorem is stronger: It states that finite groups of odd order are solvable. But it suffices to consider simple groups using a Jordan-Hölder decomposition, right? $\endgroup$
    – Martin Brandenburg
    Commented Nov 2, 2014 at 22:18
  • $\begingroup$ Well yeah, a composition of an odd group is going to have only odd quotients and since these quotients are simple of odd order they are going to be of prime order ergo abelian. so a composition series of an odd group is going to have only abelian quotients and therefore the group is solvable. $\endgroup$
    – Asinomás
    Commented Nov 2, 2014 at 22:22
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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.

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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.

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    $\begingroup$ I was pretty sure the nonexistence of radical formulas was shown by Abel. Galois' results have far more sweeping consequences but I think he wasn't the first. $\endgroup$
    – Matt Samuel
    Commented Nov 3, 2014 at 3:27
  • $\begingroup$ @MattS Wikipedia agrees with you. Thanks for the correction, fixed! $\endgroup$
    – Felipe Jacob
    Commented Nov 3, 2014 at 3:35
  • $\begingroup$ Cubic equations were solved in the 13th century, which was before the renaissance. $\endgroup$
    – Matt Samuel
    Commented Nov 3, 2014 at 3:36
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    $\begingroup$ According to wikipedia and Needham's complex analysis book the general method was found in the early 16th century. $\endgroup$
    – Felipe Jacob
    Commented Nov 3, 2014 at 3:42
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Conjectured by Euler and Legendre and proven by Gauss:

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

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    $\begingroup$ Where $p$ and $q$ are positive, distinct, odd primes. $\endgroup$
    – Dylan Yott
    Commented Nov 3, 2014 at 5:39

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