That makes it interesting to most mathematicians (if it weren’t easy to state, it could still be interesting, but to a smaller audience)
Edit: it also is easy for non-mathematicians with a knack for computing to get a crack at. You need not know much about number theory to write a program that searches for solutions for the unknown cases (it will help, but knowing how to optimize programs may help more, and that’s not something all mathematicians are good in). That grows the set of people who might find this interesting.
https://en.wikipedia.org/wiki/Sums_of_three_cubes#Computatio...
33 and 42 were known to be exceptions of all sums less than 100 for which solutions were found, until recently. 42 was the most recent to fall.
Why do we care if every integer not equal to 4 or 5 modulo 9 is the sum of three cubes? Just for fun?
In number theory you study properties/patterns of the numbers. This is an example of such a property.
These properties often seem like they don’t have any applications, and often they don’t, but not always. For example, cryptography is basically all based on number theory.
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.
As a prime example, Fermat's Last Theorem was not a particularly applicable mathematical theorem. But when Andrew Wiles found a proof, he ended up proving several other elliptic curve conjectures which were important to the field.
Was featured on HN earlier: https://news.ycombinator.com/item?id=19492091
Lot of context there.
EDIT: If you're opposed to this comment, I request that you explain your reasoning. Making the assumption that happiness is of any value, there's actually quite a bit to dig into here.
As for "hippies are pretty decent at sniffing things out", just because someone was correct about something in the past (despite a lack of evidence) does not make them correct today if they still lack evidence.
The burden of proof is on numerologists to show that they are correct, not on everyone else to disprove it.
(But I didn't bring it up, someone else did -- I was just responding to the argument that "there might be something to it".)