Timeline for hard diophantine equation: $x^3 + y^5 = z^7$
Current License: CC BY-SA 2.5
8 events
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| Feb 22, 2010 at 11:01 | comment | added | Kevin Buzzard | I took the trouble to look at the Darmon/Granville paper. It says (on the version on Darmon's web page) near the bottom of p16 "It is easy to see that the proof extends to...arbitrary number fields". They take care to formulate the statement in such a way in the number field case that units don't mess them up. | |
| Feb 21, 2010 at 20:20 | comment | added | Yaakov Baruch | Also some extra condition should be added to prevent constructing trivial sporadic solutions by using roots, maybe something like Q(x^p,y^q,z^r)=Q(x,y,z)? | |
| Feb 21, 2010 at 14:48 | comment | added | Victor Miller |
One point about the equation over number fields: the most natural way of defining a solution to be primitive would be $\min(\text{ord}_{\pi}(a),\text{ord}_{\pi}(b),\text{ord}_{\pi}(c)) = 0$, for all primes $\pi$, but then you could always multiply such a solution by suitable units and get another, so you'd need a stronger definition of "primitive".
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| Feb 21, 2010 at 14:29 | comment | added | Victor Miller | I haven't looked at the Darmon/Granville argument lately, but, from what I remember, they first show, in the hyperbolic case, that the primitive solutions must lie on a finite set of curves of genus 2, and then invoke Falting's theorem. However, I'm not sure if the first applies over general number fields. There might be some subtlety with units (i.e. if there is an infinite unit group). But, I would think that you might be able to figure this out from Beukers' paper above. | |
| Feb 21, 2010 at 12:44 | comment | added | Kevin Buzzard | As far as I know the only difference is that over a general number field the elliptic curves showing up might have positive rank, so you might get infinitely many solutions in an elliptic case. I have forgotten Darmon's argument that in the hyperbolic case the number of solutions are finite, but I do remember that he reduces to Faltings' theorem so surely the argument will work in the general number field case...(famous last words) | |
| Feb 21, 2010 at 12:32 | comment | added | Yaakov Baruch | From what little I can understand, the 2 papers above deal with the problem over Z. Can one generalize the distinction among hyperbolic (finite solution sets), euclidean (elliptic curves) and spherical (polynomial parametrization) cases to any number field? As an aside, one small "hyperbolic" example stands out in Z[i]: (-2+i)^3+(-2-i)^3=(1+i)^4. | |
| Feb 21, 2010 at 2:59 | history | edited | Victor Miller | CC BY-SA 2.5 |
added 191 characters in body
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| Feb 20, 2010 at 0:23 | history | answered | Victor Miller | CC BY-SA 2.5 |