|
Comments by Henri Cohen on the proof that the list of solutions is complete. (Start)
This is now completely standard. Cremona's mwrank program tells us that
this is an elliptic curve of rank 2 with generators P1=(-2,3) and P2=(4,9).
We now apply the algorithm (essentially due to Tzanakis and de Weger)
described in Nigel Smart's book on the algorithmic solution of Diophantine
equations: using Sinnou David's bounds on linear forms in elliptic
logarithms one finds that if P is an integral point then
P=aP1+bP2 for |a| and |b| less than a huge bound B (typically 10^100, sometimes
more, I didn't do the computation here), but the main point is that B
is completely explicit. One then uses the LLL algorithm: this is crucial.
A first application reduces the bound to 200, say, then a second application
to 20, and sometimes a third to 12 (at this point it is not necessary). Then
a very small search gives all the possible integer points. (End)
| 