Experimental Mathematics - Open Problems

Open Problems

Some relations have been shown to hold to very high precision, but no formal proof has yet been found; one example is:

\begin{align} &\sum_{n=0}^\infty \left(\frac{1}{(7n+1)^2}+\frac{1}{(7n+2)^2}-\frac{1}{(7n+3)^2}+\frac{1}{(7n+4)^2}-\frac{1}{(7n+5)^2}-\frac{1}{(7n+6)^2}\right) \\ &\stackrel{?}{=} \ \frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2} \log \left| \frac{\tan t + \sqrt{7} }{\tan t - \sqrt{7} } \right|dt, \end{align}

which has been verified to 20,000 digits.

Read more about this topic:  Experimental Mathematics

Famous quotes containing the words open and/or problems:

    “It’s fairly obvious that American education is a cultural flop. Americans are not a well-educated people culturally, and their vocational education often has to be learned all over again after they leave school and college. On the other hand, they have open quick minds and if their education has little sharp positive value, it has not the stultifying effects of a more rigid training.”
    —Raymond Chandler (1888–1959)

    “The problems of victory are more agreeable than the problems of defeat, but they are no less difficult.”
    —Winston Churchill (1874–1965)