famous open questions in mathematicsFamousAndInfamousOpenQuestionsInMathematics2004-10-09 17:57:592008-10-19 20:55:55Exampleopen question% this is the default PlanetMath preamble. as your knowledge
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% define commands hereDespite the fact that at least a generation of mathematicians has tried to solve these problems, they still remain open:
\begin{enumerate}
\item Goldbach conjecture
\item Twin prime conjecture
\item Poincar\'e conjecture
\item Riemann Hypothesis
\item Birch and Swinnerton-Dyer conjecture
\item P vs NP problem
\item Hodge conjecture
\item Solution to the Navier-Stokes equations
\item Collatz problem
\item Schanuel's conjecture
\item Beal conjecture
\item Irrationality of Euler's constant $\gamma$
\item Existence of an odd perfect number
\item Koethe conjecture
\item Invariant subspace problem
\item Lehmer's conjecture
\end{enumerate}
Open problems 3 to 8, together with the quest for a mathematical foundation explaining the mass gap property in Yang-Mills theory, constitute a collection of problems known as the Millennium Problems. Please see \PMlinkexternal{http://www.claymath.org/millennium/}{http://www.claymath.org/millennium/} for more detail.
\textbf{Note:}
A mathematical foundation for the partial solution of the the mass gap property in Yang-Mills theory has been published by G. Cleaver and K. Tanaka (2000): ``Ratio of Quark Masses in Duality Theories.'',
\PMlinkexternal{on line}{http://arxiv.org/abs/hep-th/0002089v1}, and can be accessed as a
\PMlinkexternal{PDF file}{http://arxiv.org/PS_cache/hep-th/pdf/0002/0002089v1.pdf} .