Re: Diophantic equation (chess related)

In article <d5t1ts$1gd$2@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hauke Reddmann <fc3a501@xxxxxxxxxxxxxx> wrote:
>-u**3*v**3-u**3*v+3*u**2*v**2-u**2+u*v**3-3*u*v+v**2+1=0
>
>Any solutions for rational u,v?

That's -u^3*v^3-u^3*v+3*u^2*v^2-u^2+u*v^3-3*u*v+v^2+1 = 0 , i.e.

3 3 3 2 2 2 3 2
-u v - u v + 3 u v - u + u v - 3 u v + v + 1 = 0

This describes a curve of genus 4, so there are at most finitely many
rational points. I don't know a good way to find them.

If you change variables by letting u = v*w then you will see only
even powers of v and so the curve is a 2-fold cover of the curve
-w^3*V^3+w*V^2+3*w^2*V^2-w^3*V^2+V-3*w*V-w^2*V+1 = 0 :

2 2 3 2 3 3
1 + (1 - 3 w - w ) V + (w + 3 w - w ) V - w V = 0

This one is only of genus 2. It again has only finitely many rational
points and so obviously we could answer the OP's question if we could
find them. I tried to do a similar mapping to a curves of genus 1; for
them I could hope to get a provably complete, and possibly finite, set
of rational points. But I didn't see any obvious maps to a genus-1 curve.

dave

.

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