The Perfect Cuboid Problem

Find positive integers a, b, c such that a²+b², a²+c², b²+c² and a²+b²+c² are all perfect squares — or prove none exist.

Euler studied near-misses (boxes with seven of eight requirements integer); the problem predates clean attribution. Searches rule out enormous boxes (space diagonals into the trillions with primitive constraints), and each face-diagonal condition alone defines an elliptic-curve-rich landscape — but the simultaneous system has defeated both number theory's heavy machinery and distributed computation alike. First-year-student question; research-frontier answer.