[eigenvalue]

With these longstanding open problems, it's often not the result itself that is useful so much as the new techniques or the unification of previously unrelated branches of math that are invented/discovered to prove the result. Fermat's last theorem by itself didn't really lead to much new math, but the proof of it certainly did (and showed previously unknown relationships between modular forms and elliptic curves, and opened up new areas of research in number theory, algebraic geometry, and arithmetic geometry).