Why has the Perfect cuboid problem not been solved yet?

Question

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.

Answer 1

If I were to take a guess, I'd suggest that the reason it hasn't been solved yet is because there's not any apparent practical application, and nobody's put a bounty on it that's large enough to make it worth anybody's trouble to solve.

Answer 2

It seems that the perfect cuboid problem is one approaching the difficulty of, if not more difficult than solving Fermat's Last Theorem which took Andrew Wiles many years of his career and the use of extrememly complex mathematics to do.

However, if the problem is approached logically there seems to be a good reason why one can't exist. I was never one to believe that a seemingly impossible problem should be tackled by using computer programs to try as many combinations as possible just to see if a solution could be found. To me that admits defeat and actually proves nothing unless a set of figures that proves the case is found which for a perfect cuboid hasn't happend to date.

I believe there is an answer and it's been staring us in the face but been ignored. Take the three face diagonals of a cuboid with dimensions X, Y Z. The three face diagonals are given by:-

X^2 + Y^2 = D1^2;
X^2 + Z^2 = D2^2;
Y^2 + Z^2 = D3^2;

We need go no further because all we need do is look at Y^2 and think about symmetry.