A technique for certain diophantine equations that are equivalent to asking for $x^2 - k x y + y^2 = C$ with $x,y,k$ positive integers

Vieta jumping is the observation that if $x^2 - k x y + y^2 = C$ has a solution in positive integers $x,y \geq 1,$ with $k \geq 3,$ then there is another solution with smaller $x+y$ given by replacing $x$ with $ky - x.$ Smaller, that is, unless we already have $2x \leq ky.$ Note that $C$ may be positive or negative, depends on the problem. A solution with both $0 < 2x \leq ky$ and $0 < 2y \leq kx$ is what Hurwitz called a Grundlösung in a 1907 article. Good things come from considering these fundamental solutions. The Hurwitz article is, by far, the best introduction to this technique. 1907, Über eine Aufgabe der unbestimmten Analysis. Also high proportion of formulas to text, so the German is not much of a problem. Archiv der Mathematik und Physik, volume $11,$ pages 185-196.