Uses of Vieta Jumping in research mathematics?

Question

Vieta jumping has been a prominent method for solving Diophantine equations since 1988. It was popularized when it was used to solve an IMO problem, but has it been applied to research mathematics, and applied in solving previously unsolved questions?

Context: Vieta Jumping is a method of showing that for quadratics $f$ and $g$, if two positive integers $A, B$ satisfy $f(A,B)|g(A,B)$, then there is a method to generate $A'$ such that $f(A',B)|g(A',B)$ where $A'$ is always larger/smaller than $A$. We can then continue applying this method to generate infinitely many unique pairs of numbers of the form $(x,B)$ such that $f(x,B)|g(x,B)$.

The method is as follows: Suppose $f(A',B)|g(A',B)$ and $$\frac{g(A,B)}{f(A,B)}=k$$ for some positive integer $k$. Then, this implies

$$g(A,B)-kf(A,B)=0$$

We can then rewrite this as a quadratic in terms of $A$. As this is a quadratic, there exists some $A'$ which is also a root of $g(x,B)-kf(x,B)=0$. We can then use Vieta's formulae to show that $A'$ is a positive integer, and also that $A'$ is larger/smaller than $A$.

Hence, there are infinitely many solutions to the diophantine equation $\frac{f(x,y)}{g(x,y)}=k$. If $A'\leq A$, then Vieta Jumping can be used as a form of infinite descent. If $A'\geq A$, then Vieta Jumping tells you that there are infinitely many solutions.

Though Vieta Jumping was used in the solution for Problem 6 of the 1988 International Mathematics Olympiad, it has existed before that in various other names.

Answer 1

Vieta Jumping has a much longer history than I previously thought, and in fact it's not really a new innovative method or anything, and already known in more advanced number theory as reduction theory of quadratic forms. It has been known since the times of Gauss, when he used it in Disquisitiones Arithmeticae (not exactly sure where though), and more recently by Hurwitz when analysing the Hurwitz equation. IMO students were not expected to know about reduction theory of quadratic forms pre-1988, and so when it was re-discovered in 1988 the method gained the new name of Vieta jumping.

I am interested in the application of Vieta jumping/reduction theory of quadratic forms post 1988, so feel free to post an answer if you know anything about that.

Answer 2

The technique appears as early as 1880 in Markov's work on the solutions to the Diophantine equations $$x^2 + y^2 + z^2 = 3xyz$$ now known as Markov triples. In his second article Sur les formes quadratiques binaires indéfinies he uses the technique to show that if $(x,y,z)$ is a solution, then so is $(x,y,3xy-z)$; this is equation (26) in the paper. This can be used not just to generate infinitely many solutions, but to arrange all the solutions in an infinite tree known as the Markov tree.