Does this equation have an infinite number of rational solutions: x^2+y^2=2?

My answer to a question on Quora: Does this equation have an infinite number of rational solutions: x^2+y^2=2?

I want to add, as a comment to the excellent post by Senia Sheydvasser in this thread, a simpler proof of the following statement:

If \(Ax^2 + Bxy + Cy^2 = D\) is a quadratic curve with rational coefficients \(A, B, C, D\), and if it has a point \((a,b)\) with rational coordinates, then it has infinitely many points with rational coefficients.

As Senia Sheydvasser explained, we can make change of variables \(x \leftarrow x – a\), \(y \leftarrow y\) and assume, without loss of generality, that the curve has a rational point \((0, b)\) (see the diagram below). Now pick on the \(x\)-axis a point with rational coordinates \((t,0)\) and draw the line through the points \((t, 0)\) and \((0,b)\). It is well-known that this line has equation

\(\displaystyle{\frac{x}{t} + \frac{y}{b} = 1.}\)

(the so-called “equation of a straight line in terms of its intercepts”).

The points of intersection of the line and the curve are solutions of the system of simultaneous equations

\begin{eqnarray*} \frac{x}{t} + \frac{y}{b} &=& 1\\ Ax^2 + Bxy + Cy^2 &=& D \end{eqnarray*}

Expressing \(y\) in terms of \(x\) in the first equation and substituting it into the second equation yields a quadratic equation for \(x\) with rational coefficients, something like \(Px^2 +Qx +R = 0\); one of its solutions is \(x = 0\); hence \(R = 0\) and the second root is found from a linear equation \(Px + Q = 0\) and is therefore also rational, and then the corresponding value y is also rational — it is linked to \(x\) by a linear equation with rational coefficients.

We found a way to assign to every rational point \((t,0)\) on the \(x\)-axis a point on the curve with rational coordinates.

What we did is that we used what is known as the birationality property of the Stereographic projection, one of its many wonderful properties. In a proper system of mathematics education, stereographic projections should be taught to schoolchildren who plan to seriously study, later at university, mathematics or mathematically intensive subjects in sciences or engineering. It explains a lot of stuff, say, the so-called universal trigonometric substitutions of calculus, and, in its 3-dimensional version, is critically important for understanding of complex analysis.