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.
Best Answer
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.