Quadratic Diophantine equation – I

Quadratic Diophantine equation is an equation of the form Ax**2 + Bxy + Cy**2 + Dx + Ey + F = 0 where A, B, C, D, E, and F are integer constants and x and y being integer variables. Study of this equation has always been an interesting area among number theorists. The famous pell equation is a special case of the above with delta = B**2-4AC > 0 and delta not being a perfect square. Normally, this equation is broken down into five cases for analytical purposes.

1) A = B = C = 0 (Linear case): Reduces to a linear Diophantine equation of two variables.

2) A = C = 0 and B != 0  (Simple hyperbolic case): Equation reduces to (Bx + E) (By + D) = DE – BF, which can be solved by considering the factors of DE – BF.

3) B**2 – 4AC < 0 (Elliptical case): In this case, values of the x should lie between the roots of the equation (B**2-4AC)x**2 + 2(BE – 2CD)x + E**2 – 4CF = 0. Values for x should be selected so that y is an integer.

4) B**2 – 4AC = 0 (Parabolic case): Solution procedure is rather complex in this case.

I will describe these cases in detail in the future posts. I had almost completed the above cases at the start of Week 2.

5) delta = B**2 – 4AC > 0: This is split into several subcases.

Case delta = B**2 – 4AC > 0:

Subcase 1: D = E = 0:

This is the homogeneous case and again considered under two cases F = 0 and F != 0. If F = 0, then x = 0 and y = 0 are solutions. More solutions may exist if B**2 – 4AC is a perfect square. Otherwise x = 0 and y = 0 is the only solution. I implemented this case in the module. If F != 0 the solution procedure is rather complex and involves continued fractions. I am currently working on this.

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2 Comments

  1. There’s a typo in your description of case #2. It should be “A = C = 0, B != 0″/

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