S.O.S. Mathematics CyberBoard

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Hi everyone!

I wasn't sure where to post this so I posted it here. Perhaps it should be moved to somewhere more relevant...

I'm trying to prove that no perfect cuboid exists. I have an idea how to prove it: Every Primitive Pythagorean Quadruple (PPQ) can be transformed into another using 8 transformations. Starting with (1,2,2,3), it is possible to get at all other PPQs. One of these transforms is (a,b,c,d)->(b+c+d,a+c+d,a+b+d,a+b+c-d).

The inverse transforms also exist. My idea is this: lets say we have a cuboid (a,b,c) which satisfies the perfect cuboid criteria (a^2 + b^2, a^2 + c^2, b^2 + c^2, and a^2 + b^2 + c^2 are all perfect squares, and all of a,b,c are nonzero). Is it possible to prove that the inverse transform of this cuboid is a perfect cuboid (and vice versa)? If this can be proved, it's relatively straightforward to prove that no perfect cuboid exists.

I've been sitting at the computer for the past few days, doing exhaustive searches on possible cuboids and the above theorem seems to hold, but I haven't been able to prove it.

My major barrier to cracking this problem is my feeble knowledge of number theory.
I would appreciate it if anyone can help me out, esp. in the maths concerning perfect squares.

I would also appreciate it if anyone has links to people who have done serious theoretical work on this problem (not just computer programs to search for the perfect cuboid).

Thanks in advance!

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No thoughts, anyone?