Parametrization of solutions of diophantine equation

Question

The issue I discussed in this thread. Parametrization of solutions of diophantine equation $x^2 + y^2 = z^2 + w^2$

Generally speaking at the forum often ask a question about this equation. So I think that will not solve different each time Diophantine equation is better to write the equation in this General form:

$$ax^2+bxy+cy^2=ez^2+jzw+tw^2$$

$a,b,c,e,j,t - $ integer coefficients which are defined by the problem statement.

The task is simple - to write a formula describing the parameterization of the equation. The formula itself and will specify conditions when possible integer solutions.

Many people like Diofantos geometry, but its methods are known for a very long time - here is inefficient. It is always better to have a single formula describing all equations than every time to solve the new equation.

Answer 1

Equation if we write in the General form:

$$aX^2+bXY+cY^2=eZ^2+jZW+tW^2$$

If in this equation there any equivalent to a quadratic form in which the root is an integer.

$$q=\sqrt{b^2+4a(e+j+t-c)}$$

Then there are solutions. They can be written by making the replacement.

$$x=(b(2(e+j+t)-b)+4ac)s-(b+2a)(j+2t)k$$

$$y=(b^2+4c(e+j+t-a))s^2-2(b+2c)(j+2t)sk+(j^2+4t(a+b+c-e))k^2$$

Then decisions can be recorded and they are as follows:

$$X=(b-2(e+j+t-c)\pm{q})p^2+2(q((j+2t)k-(b+2c)s)\pm{x})pn+$$

$$+(((2(e+j+t-c)-b)\pm{q})y+2((j+2t)k-(b+2c)s)x)n^2$$

$$***$$

$$Y=(\pm{q}-(b+2a))p^2+2(q((j+2t)k-(2(e+j+t-a)-b)s)\pm{x})pn+$$

$$+(((b+2a)\pm{q})y+2((j+2t)k-(2(e+j+t-a)-b)s)x)n^2$$

$$***$$

$$Z=(\pm{q}-(b+2a))p^2+2(q((j+2t)k-(b+2c)s)\pm{x})pn+$$

$$+(((b+2a)\pm{q})y+2((j+2t)k-(b+2c)s)x)n^2$$

$$***$$

$$W=(\pm{q}-(b+2a))p^2+2(q((2(a+b+c-e)-j)k-(b+2c)s)\pm{x})pn+$$

$$+(((b+2a)\pm{q})y+2((2(a+b+c-e)-j)k-(b+2c)s)x)n^2$$

$p,n,k,s $ - integers asked us.

Answer 2

If the homogeneous quadratic Diophantine equation

$$a m^2+b m n+c n^2=d p^2+e p q+f q^2$$

has a set of special solution, then a family of parametric solution can be obtained.

\begin{align*} &\qquad\quad\begin{split} a X^2+b X Y+c Y^2&=d Z^2+e W Z+f W^2\\ \end{split}\\ \\ &\left\{\begin{split} X&=-(a m+b n)r^2+c m s^2-d m u^2-f m v^2\\ &\qquad-2 c n r s+(2 d p+e q)r u+(e p+2 f q)r v-e m u v\\ Y&=a n r^2-(b m+c n)s^2-d n u^2-f n v^2\\ &\qquad-2 a m r s+(2 d p+e q)s u+(e p+2 f q)s v-e n u v\\ Z&=a p r^2+c p s^2+(d p+e q)u^2-f p v^2\\ &\qquad+b p r s-(2 a m+b n)r u-(b m+2 c n)s u+2 f q u v\\ W&=a q r^2+c q s^2-d q u^2+(e p+f q)v^2\\ &\qquad+b q r s-(2 a m+b n)r v-(b m+2 c n)s v+2 d p u v \end{split}\right. \end{align*}

   a*X^2 + b*X*Y + c*Y^2 - d*Z^2 - e*Z*W - f*W^2 /. {
   X -> -(a*m + b*n) r^2 + c*m*s^2
     - d*m*u^2 - f*m*v^2
     - 2 c*n*r*s + (2 d*p + e*q) r*u
     + (e*p + 2 f*q) r*v - e*m*u*v,
   Y -> a*n*r^2 - (b*m + c*n) s^2
     - d*n*u^2 - f*n*v^2
     - 2 a*m*r*s + (2 d*p + e*q) s*u
     + (e*p + 2 f*q) s*v - e*n*u*v,
   Z -> a*p*r^2 + c*p*s^2
     + (d*p + e*q) u^2 - f*p*v^2
     + b*p*r*s - (2 a*m + b*n) r*u
     - (b*m + 2 c*n) s*u + 2 f*q*u*v,
   W -> a*q*r^2 + c*q*s^2
     - d*q*u^2 + (e*p + f*q) v^2
     + b*q*r*s - (2 a*m + b*n) r*v
     - (b*m + 2 c*n) s*v + 2 d*p*u*v} // Factor