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Finding coprime integer solutions

asked 2011-01-20 11:45:54 -0500

ASedarous gravatar image

I'm looking to find a way to find all coprime integer solutions to an equation. As an example, the equation xy + yz + z*x == 0, which I already know the answer to. Thanks!

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Being a little more explicit about your question might help us help you. Are you looking for programming ideas? A command? What does "all coprime solutions" mean, precisely. Thanks!

kcrisman gravatar imagekcrisman ( 2011-01-24 04:31:38 -0500 )edit

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answered 2013-12-15 00:31:06 -0500

John Cremona gravatar image

Since your equation is homogeneous in 3 variables it defines a projective curve, and it is possible to search for points on it like this:

sage: P2Q.<x,y,z> = ProjectiveSpace(QQ,2)
sage: f = x*y+y*z+z*x
sage: C = Curve(f)
sage: C
Projective Conic Curve over Rational Field defined by x*y + x*z + y*z
sage: C.rational_points(bound=10)
[(-3 : -3/2 : 1),
 (-2 : -2 : 1),
 (-3/2 : -3 : 1),
 (-2/3 : 2 : 1),
 (-1/2 : 1 : 1),
 (-1/3 : 1/2 : 1),
 (0 : 0 : 1),
 (0 : 1 : 0),
 (1/2 : -1/3 : 1),
 (1 : -1/2 : 1),
 (1 : 0 : 0),
 (2 : -2/3 : 1)]

Note that the current implementation of point-finding on plane curves over QQ is very naive, but a better implementation is currently under development.

Secondly, the solutions are presented in projective coordinates normalised so that the last nonzero coordinate is 1. To get coprime integer solutions just scale up.

sage: Pts = C.rational_points(bound=10)
sage: [P.clear_denominators() for P in Pts]
[None, None, None, None, None, None, None, None, None, None, None, None]
sage: Pts
[(-6 : -3 : 2),
 (-2 : -2 : 1),
 (-3 : -6 : 2),
 (-2 : 6 : 3),
 (-1 : 2 : 2),
 (-2 : 3 : 6),
 (0 : 0 : 1),
 (0 : 1 : 0),
 (3 : -2 : 6),
 (2 : -1 : 2),
 (1 : 0 : 0),
 (6 : -2 : 3)]
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Asked: 2011-01-20 11:45:54 -0500

Seen: 665 times

Last updated: Dec 15 '13