[summary: Are problems with solutions easy to check also easy to solve? ]
The greatest currently open problem in complexity theory and computer science in general, which talks about the relationship between the class of efficiently solvable problems, [ $~$P$~$], and the class of problems with easily checkable solutions, [ $~$NP$~$].
It is a basic result that [ $~$P\subseteq NP$~$]. However, we still do not know whether the reverse implication, $~$P \supseteq NP$~$, holds.
A positive, constructive result would have profound implications through all the fields of science and math, not to mention the whole of society.