[nwallin]

Probably.

It's possible that solving this problem in a satisfactory way would teach us something useful about number theory, that's useful in real world practical stuff.

Andrew Wiles' proof of Fermat's Last Theorem is based on the modularity of elliptic curves. I'm not sure to what extent Wiles' proof contributed to elliptic curve cryptography hitting the scene a decade later, but it probably didn't hurt.