Find positive integers \(a, b, c\) such that \(a^2+b^2\), \(a^2+c^2\), \(b^2+c^2\) and \(a^2+b^2+c^2\) are all perfect squares — a box with all edges, face diagonals and the space diagonal integral — or prove none exist. An Euler brick drops the space diagonal (e.g. \((44,117,240)\)); no perfect cuboid is known, and none is ruled out.
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.
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