[GAP Forum] Re: cubic equations
Nikos Apostolakis
nikos.ap at gmail.com
Sat Feb 9 19:45:57 GMT 2008
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
>
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