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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.

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Another reason is that it is not a particular interesting problem from a mathematical perspective, its just a random diophantine system. Its truth value has probably no implications. – TROLLKILLER Apr 8 '11 at 4:43
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There aren't many implications of Fermat's Last Theorem, either. – Douglas Zare Apr 8 '11 at 5:03
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FLT is probably famous because of the little note, rather than its use. At the same time, the proof of FLT proved the Taniyama-Shimura conjecture. So it was useful – picakhu Apr 8 '11 at 5:09
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@Ganesh: I see the individual tallies (this is one of the reputation-dependent "privileges"). The two people who upvoted it must have retracted their upvotes. – joriki Apr 8 '11 at 7:16
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In the newer edition of Guy's book, he deals with Euler brick and related problems in pp. 275–283. Google books link. – Martin Sleziak Aug 5 '12 at 6:30
up vote 3 down vote accepted

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.

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Bounties and practical applications are not at all what motivate most mathematicians to attack a given problem. – Chris Eagle May 28 '12 at 19:08
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Hey, I just said it was a guess. Maybe I'm just being a bit cynical, but I sincerely think if a sizable bounty was put on this problem, I think it'd be solved (one way or the other) in under a year. – Mark May 28 '12 at 21:07
    
Since when do mathematicians, especially number theorists, care about practical applications? – anomaly Jun 20 at 20:55
    
Since the invention of RSA and elliptic curve cryptography. – Ganesh Jun 20 at 21:29
    
RSA involves only elementary mathematics, and elliptic curve cryptography is one very small slice of mathematics. There are many more interesting topics in number theory, and most mathematicians--- even in number theory--- are not in it for the practical applications. – anomaly Jun 20 at 21:59

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.

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And so the answer is.... – anomaly Jun 20 at 20:47

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.

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Indeed. For me, it's interesting because this is the kind of problem that seems to be easy - until I hear about the failure of past attacks. – Ganesh Jun 20 at 21:00
    
But any sort of Diophantine equation is hard to solve; this particular one just doesn't have any interesting connections with number theory, algebraic geometry, etc. – anomaly Jun 20 at 21:03
    
True, but it has geometric roots. – Ganesh Jun 20 at 21:30
    
Not really. The geometry involved is trivial and not particularly compelling. – anomaly Jun 20 at 21:34
    
Agreed. I merely pointed out that the problem has geometric roots, unlike a random diphantine system. – Ganesh Jun 20 at 21:47

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