When is $x^2+y^2+1=xyz$?

Question

Essentially, when is $\dfrac{x^2 +y^2+1}{xy}$ a positive integer?

I've tried many approaches such as considering the above equation as a quadratic in $x$ or $y$ but I haven't had much success. The only solutions I could find were $x=1$, $y=1$ and $z=3$.

A similar problem: I know that the only solutions to $\dfrac{x^2+y^2}{xy}$ to equal a positive integer is for $x$ and $y$ to both be one. This is pretty much considering when a number, not necessarily an integer, and it's inverse sum to make a positive whole number. However, the problem is a slight variation, it asks when a number $\frac{p}{q}$ summed with it's inverse $\frac{q}{p}$ plus $\frac{1}{pq}$ is a whole number.

"giving you all solutions" is a bit of an overstatement, at lest there is work to do, see my answer. But your observation is crucial. +1 — A. Pongrácz, Aug 15 '18 at 18:49
@A.Pongrácz You are right, it is more of a first step towards all solutions. — Sil, Aug 15 '18 at 18:58
Also quite interestingly, as one of the comments in linked questions suggests, it is linked to Fibonacci numbers, since $F_n^2+F_{n+2}^2 + 1= 3 F_{n}F_{n+2}$ for $n$ odd. — Sil, Aug 15 '18 at 19:07

A. Pongrácz · Accepted Answer · 2018-08-15T18:48:34.987

1

Using this post, suggested in the comments: Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.

we have $z=3$.

$x^2 - 3y\cdot x + (y^2+1)=0$, so the discriminant is a perfect square: $D=5y^2-4=t^2$. This is a Pell-type equation with infinitely many solutions: e.g., $y=2, t=4$, or $y=5, t=11$.

https://en.wikipedia.org/wiki/Pell%27s_equation

edited Aug 15 '18 at 18:48

answered Aug 15 '18 at 18:41

A. Pongrácz

7,418

Thanks so much, you're the best – Dean Yang Aug 15 '18 at 18:56
the best presentation related to "Vieta Jumping" is Hurwitz(1907) and the German is on the easy side; http://zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf If curious, you might view the roughly 45 questions with tag https://math.stackexchange.com/questions/tagged/vieta-jumping – Will Jagy Aug 15 '18 at 19:29

When is $x^2+y^2+1=xyz$?

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