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:

    “All deep, earnest thinking is but the intrepid effort of the soul to keep the open independence of her sea; while the wildest winds of heaven and earth conspire to cast her on the treacherous, slavish shore.”
    —Herman Melville (1819–1891)

    “If combat means living in a ditch, females have biological problems staying in a ditch for 30 days because they get infections.... Males are biologically driven to go out and hunt giraffes.”
    —Newt Gingrich (b. 1943)