There can be very long periods where number theory advances are no more useful than say string theory or literary theory. Then every once in a while you hit something big. Much of encryption rests on what was formerly ivory tower papers on prime numbers.

This isn't to increase or diminish the importance of the accomplishment. I actually don't know where this belongs in the annals of math.

We are one step closer to immortality.

If finding such a numbers is much more difficult than checking if a number satisfies the requirements then it could be used in cryptography as an alternative to finding prime numbers or generating hashes. Although I am not sure there are enough of these numbers for any arbitrary number of squares to make it practical.

Good question, and I am also curious. I don't see how this can be used for cryptography on a large scale or how it can help us in other ways.

Take Fermat's Last Theorem for example... how has that been used to improve or change life?