Uses of Vieta Jumping in research mathematics? 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.

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.
