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John R Ramsden
John R Ramsden
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Looks OK to me (the approach anyway - I didn't check the numerator calculation). Ruslan Sharipov also found a simpleran explicit equation for the perfect cuboid surface, in a recent ArXiv paper at http://arxiv.org/abs/1104.1716. But hisHis derivation was much more intricate than yours., but the result looks very similar!

This surface is known to be a so-called surface of general type [ http://en.wikipedia.org/wiki/Surface_of_general_type ] and thus has only a finite number of rational points.

Most reckon it has no non-trivial (i.e. with all non-zero) rational points, and with equations like this it tends to be "small or nothing". Various people over the last century or so have claimed proofs of this; but I think the problem is still generally agreed to be open.

It would be interesting to look at congruence conditions on a homogenized version of your equation or Sharipov's. Maybe you would find the high degree strongly limited the number of solutions modulo smallish primes such as 17 and 23, although with four variables kicking around (in the homogenized equation) there are a lot of combinations!

Looks OK to me (the approach anyway - I didn't check the numerator calculation). Ruslan Sharipov found a simpler equation for the perfect cuboid surface, in a recent ArXiv paper at http://arxiv.org/abs/1104.1716. But his derivation was much more intricate than yours.

This surface is known to be a so-called surface of general type [ http://en.wikipedia.org/wiki/Surface_of_general_type ] and thus has only a finite number of rational points.

Most reckon it has no non-trivial (i.e. with all non-zero) rational points, and with equations like this it tends to be "small or nothing". Various people over the last century or so have claimed proofs of this; but I think the problem is still generally agreed to be open.

It would be interesting to look at congruence conditions on a homogenized version of your equation or Sharipov's. Maybe you would find the high degree strongly limited the number of solutions modulo smallish primes such as 17 and 23, although with four variables kicking around (in the homogenized equation) there are a lot of combinations!

Looks OK to me (the approach anyway - I didn't check the numerator calculation). Ruslan Sharipov also found an explicit equation for the perfect cuboid surface, in a recent ArXiv paper at http://arxiv.org/abs/1104.1716. His derivation was much more intricate than yours, but the result looks very similar!

This surface is known to be a so-called surface of general type [ http://en.wikipedia.org/wiki/Surface_of_general_type ] and thus has only a finite number of rational points.

Most reckon it has no non-trivial (i.e. with all non-zero) rational points, and with equations like this it tends to be "small or nothing". Various people over the last century or so have claimed proofs of this; but I think the problem is still generally agreed to be open.

It would be interesting to look at congruence conditions on a homogenized version of your equation or Sharipov's. Maybe you would find the high degree strongly limited the number of solutions modulo smallish primes such as 17 and 23, although with four variables kicking around (in the homogenized equation) there are a lot of combinations!

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John R Ramsden
John R Ramsden
  • 1.5k
  • 13
  • 20

Looks OK to me (the approach anyway - I didn't check the numerator calculation). Ruslan Sharipov found a simpler equation for the perfect cuboid surface, in a recent ArXiv paper at http://arxiv.org/abs/1104.1716. But his derivation was much more intricate than yours.

This surface is known to be a so-called surface of general type [ http://en.wikipedia.org/wiki/Surface_of_general_type ] and thus has only a finite number of rational points.

Most reckon it has no non-trivial (i.e. with all non-zero) rational points, and with equations like this it tends to be "small or nothing". Various people over the last century or so have claimed proofs of this; but I think the problem is still generally agreed to be open.

It would be interesting to look at congruence conditions on a homogenized version of your equation or Sharipov's. Maybe you would find the high degree strongly limited the number of solutions modulo smallish primes such as 17 and 23, although with four variables kicking around (in the homogenized equation) there are a lot of combinations!