TITLE: Why has the Perfect cuboid problem not been solved yet?
QUESTION [13 upvotes]: Why hasn't   Perfect Cuboid  Problem  been solved yet, whereas (possibly) more nontrivial ones such as FLT and Sphere packing have been solved?
I understand that calling some problems more nontrivial may be naive and seemingly trivial problems can be deceptively tricky, as with the FLT. All the same, FLT and to a lesser extent, Sphere Packing, garnered lots of attention by successive generations of mathematicians, until someone decided to finish it off and succeeded. 
But, AFAIK, the Perfect Cuboid (PC) problem hasn't generated this kind of attention, perhaps because Fermat didn't leave a note about it. Is that the reason for PC remaining unsolved? One of the standard references for PC ,  Unsolved Problems in Number Theory , suggests several numerical results (p.178), but of course nothing like a proof, much like the status of  FLT and Sphere Packing many decades ago.

REPLY [2 votes]: It's just not a particularly interesting problem. Apart from the romantic (and ridiculous) idea that Fermat had a secret proof of his conjecture that was lost to history, there's not much compelling about FLT aside from the fact that it's very easy to state. What makes it interesting is that Frey proved that given a nontrivial rational point on the curve $x^n + y^n = z^n$ with $n > 2$, he could construct a elliptic curve $E/\mathbb{Q}$ that isn't modular. That would be significant; it ties into Taniyama-Shimura, the Hasse-Weil conjecture, the Langlands program, and so on. Without it, FLT would just be another arbitrary Diophantine equation with mild historical interest, relegated to amateur and recreational math. The brick problem is not as elegant as FLT and doesn't seem to tie into anything more significant, so it's not a topic of ongoing research.