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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.

Kyan Cheung
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  • I (and many others) used more or less a similar technique to solve Problem 3 at IMO 1981. I just called it induction. I referred to that problem on this site as well, and Bill Dubuque pointed out that it is also Vieta jumping. Anyway, it is, in my opinion, a stretch to say that it was invented at IMO '88 :-) Mind you, the 1988 problem you refer to is cute. A colleague asked me about it at a party in '95 or thereabouts. I had to spend a couple of days at it (with a bit of testing with the computer) until I figured it out :-) – Jyrki Lahtonen Apr 22 '21 at 14:41
  • But, I make no claims about having researched the matter. I guess you would need a precise definition of Vieta jumping before you could discuss what was the first instance. – Jyrki Lahtonen Apr 22 '21 at 14:47
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    Well, I learnt Vieta jumping as a form of infinite descent, but I suppose I can call that Vieta jumping. It also seems that Vieta jumping has existed since 1907. And I do agree that the explanation of Vieta jumping can be worked on. – Kyan Cheung Apr 22 '21 at 15:06
  • Wait, my link just answered my question. It literally provided an example of Vieta jumping in a paper. – Kyan Cheung Apr 22 '21 at 15:29
  • A good catch @Kyky! I went for a walk and came back thinking that I need a way to summon Will Jagy here to comment on representing numbers with quadratic forms. Pell equations and such. – Jyrki Lahtonen Apr 22 '21 at 15:36
  • FWIW the IMO problem 1981/3 has the quadratic form $$f(m,n)=m^2-mn-n^2$$ and the solution relies on the symmetry $$f(m+n,m)=-f(m,n).$$ I suppose you can argue that the sign change takes it outside of the realm of Vieta jumping. No matter, it is just infinite descent/recursion/induction/whateveryouwanttocallit :-) – Jyrki Lahtonen Apr 22 '21 at 19:01
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    @JyrkiLahtonen I think I would emphasize the integer automorphism group of a quadratic form for this OP; jumping is a special case, which has an attractive expression as a single-variable, um, thing. These lead to large finite simple groups. book Thomas M. Thompson: Ffom error correcting codes through sphere packings to simple groups also, by Wolfgang Ebeling, Lattices and Codes, which was my main source for my part of http://alpha.math.uga.edu/~pete/Clark_Jagy_11_13_2013.pdf – Will Jagy Apr 22 '21 at 19:38

2 Answers2

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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.

Kyan Cheung
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    Reduction theory of quadratic forms is only one of many ways to view these results (and not always the most convenient or insightful). As I mention in this answer, these results are all special cases of old results about symmetry groups of integer points on conics. Further the group laws on conics can be viewed as special cases of the group law on elliptic curves. This beautiful geometric viewpoint leads to much unification and insight. Follow the link for literature references. – Bill Dubuque Apr 22 '21 at 17:12
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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.

Misha Lavrov
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