Computer search has confirmed the equation solvable for every n up to 10^17, and modular identities cover infinitely many residue classes, but no argument closes the remaining, conjecturally empty, set of exceptions for every n at once.
What would resolve this A general proof, or a genuine counterexample, covering every integer n rather than a further extension of the verified range.
OpenNumber theoryErdos-Straus Conjecture (Wikipedia)