I think its the manybody schroedinger equation for a superconductor.
We still havent found a way to deal with it, even approximately, and so many brilliant Nobel prize winners have tried for decades.
Its a problems that is complex (it has a lots of relevant details n it), difficult (nobody has a clue on how to even start tackling it), and has immense promise (if we could solve it and understand superconductors it would revolutionise the electronics industry, and many others). It has to take the cake.
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