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:

    “I open with a clock striking, to beget an awful attention in the audience—it also marks the time, which is four o’clock in the morning, and saves a description of the rising sun, and a great deal about gilding the eastern hemisphere.”
    —Richard Brinsley Sheridan (1751–1816)

    “The truth of the thoughts that are here set forth seems to me unassailable and definitive. I therefore believe myself to have found, on all essential points, the final solution of the problems. And if I am not mistaken in this belief, then the second thing in which the value of this work consists is that it shows how little is achieved when these problems are solved.”
    —Ludwig Wittgenstein (1889–1951)