On Sat, Feb 09, 2008 at 08:11:22AM -0800, thus spake muniru asiru: > Dear Forum, > > Please assist me in programming Gap to find x(rational > number) and y(integer number) so that > (y-1)x^3+yx^2+(y+1)x-y=0, y<>1. > > The only solutions I got is (x,y)=(1/2,3). Could > anyone help find others? >
Two more solutions are (1,0) and (56/103, - 418488) if there is an other solution with x > 0 the denominator of x is larger than 1000. I used maxima to symbolicaly solve the equation to get that the general solution is [y = (x^3-x)/(x^3 + x^2 + x -1)] then still using maxima we substitute x = m/n to get: y = (m*n^2 - m^3)/(n^3 - m*n^2 - m^2*n - m^3) Then using Gap (since this is Gap forum ;)) I did: Filtered(Cartesian([1..1000], [1..1000]), x -> IsInt((x[1]*x[2]^2-x[1]^3)/(x[2]^3-x[1]*x[2]^2-x[1]^2*x[2]-x[1]^3)));; List(last, x -> x[1]/x[2]);; Collected(last); #==> [ [ 1/2, 500 ], [ 56/103, 9 ], [ 1, 1000 ] ] Of course the above is a quick"brute force" approach. Perhaps there is a more elegant way. HTH, Nikos > Thanks > > maasiru > _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
