famous open questions in mathematics

Despite the fact that at least a generation of mathematicians has tried to solve these problems, they still remain open:

1. Goldbach conjecture

2. Twin prime conjecture

3. Poincaré conjecture

4. Riemann Hypothesis

5. Birch and Swinnerton-Dyer conjecture

6. P vs NP problem

7. Hodge conjecture

8. Solution to the Navier-Stokes equations

9. Collatz problem

10. Schanuel’s conjecture

11. Beal conjecture

12. Irrationality of Euler’s constant $\gamma$

13. Existence of an odd perfect number

14. Koethe conjecture

15. Invariant subspace problem

16. Lehmer’s conjecture

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 http://www.claymath.org/millennium/http://www.claymath.org/millennium/ for more detail.

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.”,http://arxiv.org/abs/hep-th/0002089v1on line, and can be accessed as ahttp://arxiv.org/PS_cache/hep-th/pdf/0002/0002089v1.pdfPDF file .