# Questions tagged [vieta-jumping]

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

52 questions
86 views

### Understanding a proof that, if $xy$ divides $x^2+y^2+1$ for positive integers $x$ and $y$, then $x^2+y^2+1=3xy$

This is a Worked Example from Brilliant.org's entry on Vieta Root Jumping. Let $x$ and $y$ be positive integers such that $xy$ divides $x^2+y^2+1$. Prove that $$x^2+y^2+1=3xy$$ The solution ...
40 views

### Showing that $(1, F_{2k-1}, F_{2k+1})$ is a Markoff triple for each integer $k \geq 1$

Problem: Let $(F_n)_{n\geq1}$ denote the Fibonacci sequence with $F_1 = F_2 = 1$. Show that $(1, F_{2k-1}, F_{2k+1})$ is a Markoff triple for each integer $k \geq 1$. For reference a Markoff triple ...
169 views

### Show that if $a,b,c\in \mathbb{N}$ and ${a^2+b^2+c^2}\over{abc+1}$ is an integer it is the sum of two nonzero squares

I was reading about the fascinating problem in the IMO ($1988$ #$6$) that asks: Let $a$ and $b$ be positive integers such that $(ab+1) | (a^2+b^2)$. Show that ${a^2+b^2}\over{ab+1}$ is a perfect ...
93 views

### Prime solutions to a congruence modulo a semi-prime

Let $p$ and $q$ be primes. Besides $\{3,13\}$ and $\{13,61\}$, find other solutions $\{p,q\}$ to the congruence $$1+ p+q+p^2+q^2 \equiv 0 \pmod {pq}$$ or show that there are none.
163 views

### $m+n+p-1=2\sqrt{mnp}$ in positive integers

If $m,n,p \in N$ and $m+n+p-1=2\sqrt{mnp}$, prove that at least one of $m,n,p$ is a perfect square. There is a duplicate here: Perfect square : $m+n+p-2\sqrt{mnp}=1$, but I am very confused. Now I ...
139 views

### Show that infinitely many positive integer pairs $(m,n)$ exist s.t $\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{N}$ [duplicate]

Show that infinitely many positive integer pairs $(m,n)$ exist such that $\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{N}$. I couldn't solve it but I did make an observation which might or might not be ...