Solving Quadratic Diophantine Equations with A=0, C=0

[sparsh12]

A quadratic diophantine equation is of form:

Ax^2 + Bxy + Cy^2 +Dx + Ey + F =0

Now, for A=0 and C=0,

Bxy + Dx + Ey + F=0 ...(1)

moreover there is one more condition, gcd(B,D,E)=1

So how do I find if some integral solution of (1) exists or not?
I am not interested in the solution itself, but rather just it's existence.

And the method must not depend on searching, as in the image here:
http://s9.postimage.org/dv30vaixb/diop.png

Original website was:
http://www.alpertron.com.ar/METHODS.HTM#SHyperb

Thanks in advance for advice and ideas.

 

[sparsh12]

Why no reply yet?

 

[DonAntonio]

Well, it's very hard to know what EXACTLY you want, as you say that you want to know about the existence of an integral solution "without searching" (??), but in many instance one HAS to divide the problem in cases and check each, something you apparently don't want to do...

 

[coolul007]

The equation is the standard equation for all conic sections in 2 dimensions. It seems very plausible that there are circles/parabolas/hyperbolas that have integer roots.

 

[blue_raver22]

I would approach this problem by first understanding the concept of diophantine equations and their solutions. A diophantine equation is an equation where the solutions are required to be integers. In the case of a quadratic diophantine equation, the solutions are required to be integers for both x and y.

In the given equation (1), we can see that A=0 and C=0, which simplifies the equation to Bxy + Dx + Ey + F=0. This means that there is no x^2 or y^2 term in the equation. This also means that the equation is now linear in nature, rather than quadratic. This can provide some insights into the solution of the equation.

The condition gcd(B,D,E)=1 means that the greatest common divisor of the coefficients B, D, and E is 1. This is a crucial condition for the existence of integral solutions. If this condition is not satisfied, it means that there is a common factor among the coefficients, which would result in a fractional or non-integral solution.

Now, to determine the existence of integral solutions, we can use a number of techniques such as the Euclidean algorithm or modular arithmetic. These methods do not involve searching for solutions and are based on mathematical principles and concepts. They can help us determine if a solution exists or not, without actually finding the solution itself.

For example, using the Euclidean algorithm, we can find the greatest common divisor of the coefficients B, D, and E. If this value is not equal to 1, then there are no integral solutions to the equation. On the other hand, if the value is 1, then there exists at least one integral solution.

In conclusion, as a scientist, I would approach this problem by understanding the concept of diophantine equations, using mathematical techniques to determine the existence of integral solutions, and avoiding any searching methods. This approach would ensure a more systematic and rigorous solution to the problem.