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This is the seventh most popular easily understood unsolved problem on MathOverflow. I like it because it shows there are still fairly simple identities we can check using a computer, but can't prove. These identities aren't usually important in themselves! They're just hints that there are branches of math waiting to be better understood.
Ramanujan discovered a lot of tricky identities like this. Many have been proved using further advances in modular forms - a big link between complex analysis and number theory - and also "mock modular forms", a generalization Ramanujan invented in his last letter to Hardy and his "lost notebook". (His lost notebook was found in a box in Trinity College in 1976.)
Can this identity be proved using ideas about modular forms, or mock modular forms? Or is it hinting at a whole new branch of mathematics? Only time will tell!
The whole list of unsolved problems is fun:
