I hope it is not bad form to reply to one's own replies, but I could not resist.
In complex form the equation to be solved is |z(z-1)| = 3. Substituting z=x+yi and multiplying out (this is known as "restriction of scalars") one gets the equation (x^2+y^2)((x-1)^2+y^2)=9 which, believe it or not, is an elliptic curve. Strictly, it is a smooth affine plane quartic curve but it has singularities at infinity (double points at (i:1:0) and (-i:1:0)) which bring its genus down to 1. The curve has no rational points (i.e. there are no solutions with x,y rational), in fact it has no 2-adic points. (I used Magma's IsLocallySoluble() for that). However over Q(i) one can use the point at infinity to bring the curve into Weierstrass form, where it becomes (in new variables) y^2+xy=x^3-24x+63, which has CremonaReference 429b1 (and rank 1 so infinitely many rational points). You can't get away from elliptic curves! John Cremona -- Prof. J. E. Cremona | University of Nottingham | Tel.: +44-115-9514920 School of Mathematical Sciences | Fax: +44-115-9514951 University Park | Email: [EMAIL PROTECTED] Nottingham NG7 2RD, UK | This message has been checked for viruses but the contents of an attachment may still contain software viruses, which could damage your computer system: you are advised to perform your own checks. Email communications with the University of Nottingham may be monitored as permitted by UK legislation. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-forum URLs: http://sage.math.washington.edu/sage/ and http://sage.scipy.org/sage/ -~----------~----~----~----~------~----~------~--~---
