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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

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2answers
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 ...
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1answer
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 ...
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2answers
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 ...
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1answer
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.
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2answers
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 ...
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2answers
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 ...
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4answers
228 views

Integer points on a hyperbola

Why I'm here I have the following problem in probability from a book: You have a bag with red and white balls and you draw two balls without replacing. If the probability of drawing 2 white balls ...
4
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4answers
154 views

Prove there is no $x, y \in \mathbb Z^+ \text{ satisfying } \frac{x}{y} +\frac{y+1}{x}=4$

Prove that there is no $x, y \in \mathbb Z^+$ satisfying $$\frac{x}{y} +\frac{y+1}{x}=4$$ I solved it as follows but I seek better or quicker way: $\text{ Assume }x, y \in \mathbb Z^+\\ 1+\frac{y+...
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3answers
101 views

What are the solutions of the equation $3np+3n+2=n^2+p^2$, with n and p positive integers?

Trying to find the solutions of $3np+3n+2=n^2+p^2$ with n and p positive integers, I found out n=14 and p=4 is a solution, but I want to know if there are others.
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1answer
95 views

Is the famous problem number #6 solvable in first order Peano arithmetic?

I just came accross the famous "very difficult" problem 6 of the 1988 International Mathematical Olympiad: Let $a$ and $b$ be positive integers and $k=\frac{a^2+b^2}{1+ab}$. Show that if $k$ is an ...
0
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1answer
81 views

Find all possible value of c

Let a and b be non-negative integers. If $\frac{a^2+ab+b^2}{ab-2} = c$ for some non-negative integer c, find all possible values of c. Can somebody give me some hint or tips on how to solve this? I ...
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4answers
82 views

Equation with integers $x$, $y$

If $x$, $y$ positive integers ($x<y$), how can I solve the equation $x+y=14\sqrt{xy-48}$ ?
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5answers
193 views

How prove infinitely many postive integers triples $(x,y,z)$ such $(x+y+z)^2+2(x+y+z)=5(xy+yz+zx)$

show that there exsit infinitely many postive integers triples $(x,y,z)$ such $$(x+y+z)^2+2(x+y+z)=5(xy+yz+zx)$$ May try it is clear $(x,y,z)=(1,1,1)$ is one solution,and $$(x+y+z+1)^2=5(...
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1answer
161 views

A conjecture inspired by a famous contest problem

There is a famous difficult problem: For all natural numbers $a,b$ it's true that: $\displaystyle(ab+1)|(a^2+b^2)\implies \frac{a^2+b^2}{ab+1}$ is a perfect square. I've noticed $(a=1,\dots 100)$ ...
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2answers
258 views

Let $x$, $y$ and $z$ be natural numbers satisfying $x^2 + y^2 + 1 = xyz$. Prove that $z = 3$. [duplicate]

Let $x$, $y$ and $z$ be natural numbers satisfying $x^2 + y^2 + 1 = xyz$. Prove that $z = 3$. I have managed to show that $z$ is a multiple of $3$ by looking at it modulo $3$ but not sure how to show ...
8
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1answer
253 views

Finding all primes $p,q$ with $p^2+q^2=9pq-13$

The last month I was trying to solve a problem of a magazine, and I found the following equation $$p^2+q^2=9pq-13,$$ Where $p$ and $q$ are primes. We need to get solutions when $p$ and $q$ are odd, ...
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1answer
239 views

History of Vieta Jumping method

I know the Vieta Jumping method started to be used in Mathematics Olympiads in the IMO of 1988. But I'd like to know the history of this method before being used in mathematics olympiads. Where did it ...
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4answers
848 views

Showing that $m^2-n^2+1$ is a square

Prove that if $m,n$ are odd integers such that $m^2-n^2+1$ divides $n^2-1$ then $m^2-n^2+1$ is a square number. I know that a solution can be obtained from Vieta jumping, but it seems very different ...
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1answer
2k views

Vieta Jumping: Related to IMO problem 6, 1988: If $ab + 1$ divides $a^2 + b^2$ then $ab + 1$ cannot be a perfect square.

The famous IMO problem 6 states that if $a,b$ are positive integers, such that $ab + 1$ divides $a^2 + b^2$, then $\frac{ a^2 + b^2}{ab + 1 }$ is a perfect square, namely, $gcd(a,b)^2$. How about a ...
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3answers
1k views

Equation with Vieta Jumping: $(x+y+z)^2=nxyz$.

Find all $n\in\mathbb{N}$ so that there exist $x,y,z\in \mathbb{N}$ that solve: $$(x+y+z)^2=nxyz$$ I tried to attack it finding solutions, but all the solutions doesn't seem to have something in ...
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0answers
794 views

Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Show that $(a^2 + b^2)/(ab + 1)$ is a perfect square. [duplicate]

Hi I have been working on this problem for a while and don't understand the solution given. Here is the solution: Suppose that $(a^2 + b^2)/(ab + 1) = k$ and we assume that there exists one or more ...
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3answers
210 views

Prove that, if a and b are positive integers and $ab\mid a^2 + b^2$ then $a=b$

Hi I have been working on this problem, and I don't understand the solution. Here is the problem: If $a,b$ are positive integers and $ab\mid a^2+b^2$ then $a=b$. Solution: Suppose there are pairs $(...
32
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1answer
6k views

Math olympiad 1988 problem 6: canonical solution (2) without Vieta jumping

There is a recent question about this famous problem from 1988 on this forum, but I'm unable to respond to this because the subject is closed for me (insufficient reputation). Therefore this new post ...
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1answer
146 views

Solving $(2^k-3)b^2 - (2^{k+1}+1)ab - (2^k+2)a^2 = 0$

I’m trying to solve the Diophantine equation \begin{align} \tag{$\star$} (2^k-3)b^2 - (2^{k+1}+1)ab - (2^k+2)a^2 = 0, \end{align} where $k$ is a positive integer, and $a$ and $b$ are relatively prime ...
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1answer
60 views

Why do we need the minimum solution of (A, B) with respect to some funtion in Vieta jumping?

I guess the title is self-explanatory, but according to Wikipedia, the second step is to take the minimal solution (A, B): The minimal solution (A, B) with respect to some function of A and B, ...
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2answers
107 views

Proving $ \left\{\frac{y^2+z^2+2}{yz}\ \mid\ y,z\in\mathbb{N}\right\}\cap\mathbb{N}=\{4\} $

I conjecture that: $$ \left\{\frac{y^2+z^2+2}{yz}\ \mid\ y,z\in\mathbb{N}\right\}\cap\mathbb{N}=\{4\} $$ Is this true? If yes, how to prove it?
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5answers
396 views

Find a positive integer solution to $xyzw=504(x^2+y^2+z^2+w^2)$

Find positive integer values of $x,y,z,w$, such that $$xyzw=504(x^2+y^2+z^2+w^2)$$ I found it at some point and now I am unable to find the solution anymore, maybe this equation isn't satisfiable? ...
12
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1answer
471 views

For which integers $a,b$ does $ab-1$ divide $a^3+1$?

A problem I wasn't able to solve: For which values of $a,b\in\mathbb{Z}$ does $ab-1$ divide $a^3+1$? I am looking for every possible solution. Some of them are trivial, like $a=0,b=0$ or $(a,b)\in\...
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1answer
159 views

Find the integer values of c

Find all possible positive integer values of c that $\frac {a^2+b^2+ab} {ab-1}$=c can take in $\mathbb N$. I know that the solutions are c=7 and c=4 but I don't know how to prove this with Vieta ...
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0answers
242 views

Vieta Jumping and Hurwitz 1907

Today I proved finiteness for the problem here: Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers? namely: IF we ...
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2answers
111 views

Find the postive integers such $xy+x+y\mid x^2+y^2-2$

Let $x,y$ be positive integers, and such $$xy+x+y\mid x^2+y^2-2$$ Find the $x,y$ $$x^2+y^2-2=A(xy+x+y)\Longrightarrow x^2-(Ay+A)x+y^2-Ay-2=0$$
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4answers
510 views

Diophantine equation $(x+y)(x+y+1) - kxy = 0$

The following came up in my solution to this question, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation $$ (x+y)(x+y+1) - kxy = 0$$ For $k=5$ ...
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votes
4answers
638 views

Proving that all terms in sequence are positive integers (Recursive)

I am preparing for Putnam and working through practice-like questions and I am boggled. I've got the sequence defined by $a_1 = 1$ and $a_{n+1} = 2a_n+\sqrt{3a_n^2-2}$ for any $n\in\mathbb{N}$ and I'm ...
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2answers
189 views

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn+1$ is a divisor of $d^2 + n^2$

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn+1$ is a divisor of $d^2 + n^2$. So i formed the equation that $$\frac{n}{d} = \frac{d^2 + n^2}{dn + 1}$$ And ...
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1answer
221 views

Integer solutions to the equation $a_1^2+\cdots +a_n^2=a_1\cdots a_n$

What is the general solution to the equation $$\sum_{j=1}^n a_j^2=\prod_{j=1}^n a_j,$$ $n\in \mathbb N$ , $n \ge 2$ over $\mathbb N_0$ ? WLOG, we can assume $0\le a_1 \le a_2\le \cdots \le a_n$ For ...
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3answers
219 views

Is it possible to generate a solution for this? $p^2 + q^2 + r^2 = pqr $

I have seen this problem in a local magazine. I want to Generate an formula for this for the solutions of this problem Oh and it's given that $$p,q,r \in\mathbb{N} $$ and they can be equal.
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4answers
324 views

Solve: $ab+bc+ca\mid (a+b+c)^2$

I couldn't make any progress on this problem, can anyone help? I found it's the same as: Find all integers $a,b,c$ such that $ab+bc+ca$ divides $a^2+b^2+c^2$. I found a solution $a=-b=1$, and $c$ ...
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votes
2answers
248 views

If $\frac{a+1}{b}+\frac{b}{a}$ is an integer then it is $3$.

If $\frac{a+1}{b}+\frac{b}{a}$ is an integer for positive integers $a,b$ then prove that this integer is $3$. I reduced the to prove that if $\frac{c^2+d^2+1}{cd}$ is an integer then it is $3$ where $...
13
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1answer
271 views

Integer solutions to $\prod\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}x_i^2$

Given integers $x_1,\dots,x_n>1$. Let's assume WLOG that ${x_1}\leq\ldots\leq{x_n}$. I want to prove that the only integer solutions to any equation of this type are: $x_{1,2,3 }=3\implies\prod\...
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votes
5answers
30k views

Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?

Prove or disprove that if $t$ is a positive integer, $$f(x,y)=\dfrac{x^2+y^2}{xy-t},$$ then $f(x,y)$ has only finitely many distinct positive integer values with $x,y$ positive integers. In other ...
2
votes
2answers
133 views

$ ab-1|a^2+ab+b^2 $

I hava a number theory problem. I think on it yestarday night and today, afternoon. The problem : $ a,b $ are two natural numbers such that : $ ab>1 $ how many pairs $ (a,b) $ is there such that ...
5
votes
1answer
543 views

Positive integer solutions to $x^2+y^2+x+y+1=xyz$

The question asks for positive integer solutions to $x^2+y^2+x+y+1=xyz$ . We at first note that $x|y^2+y+1$. Now,let there exist positive integers $x,y$ that satisfy the given equation.Then $mx=y^2+y+...
2
votes
1answer
138 views

Diophantine quartic equation in four variables, part deux

A recent Question asked for all positive integer solutions of a simple quartic in four unknowns: $$ wxyz = (w+x+y+z)^2 \tag{1}$$ whose satisfaction is necessary for the integer side lengths $a,b,c,d$...
4
votes
6answers
932 views

Diophantine quartic equation in four variables

Comments from a recent Question, Cyclic quadrilateral with equal area and perimeter, ask about such cases with (positive) integer lengths. Using Brahmagupta's formula for the area of a cyclic ...
13
votes
9answers
13k views

IMO 1988, problem 6

In 1988, IMO presented a problem, to prove that $k$ must be a square if $a^2+b^2=k(1+ab)$, for positive integers $a$, $b$ and $k$. I am wondering about the solutions, not obvious from the proof. ...
4
votes
1answer
159 views

General question about 'vieta jumping'

Suppose I want to prove that a variable posesses a certain property (e.g. is a square). For example if I wanted to prove that $x$ in $\frac{x^2+y^2+1}{xy} = k$ has the property of being a square (It ...
15
votes
2answers
286 views

How to prove that $a_{n}$ must be of the form $a^2+b^2$?

let $a_{1}=1,a_{2}=2,a_{3}=5$,and $$a_{n}=3a_{n-1}a_{n-2}-a_{n-3}$$ show that $a_{n}=a^2+b^2,a,b\in N$ while $a_{1}=0^2+1^2,a_{2}=2=1^2+1^2,a_{3}=5=2^2+1^2,a_{4}=29=5^2+2^2,a_{5}=433=17^2+12^2$ and ...
6
votes
2answers
1k views

Divisibility - Math Olympiad: $x|(y^2+m)$ and $y|(x^2+m) $

Show that for any positive integer $m$, there is an infinite number of pairs of integers $(x,y)$ satisfying the conditions: i) $\gcd(x,y)=1 $; ii) $y \mid x^2+m$; iii) $x \mid y^2+m$.
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2answers
1k views

Resource for Vieta root jumping

I can't seem to find a good resource on Vieta's root jumping, which would explain various scenarios that suggest using the technique. Does anyone have a suggestion for a reference?
1
vote
4answers
404 views

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$.

So I came up with $b= a+1$ $\Rightarrow$ $ab=a(a+1) = a^2 + a$ So that: $a^2+b^2 -1$ = $a^2 + (a+1)^2 -1$ = $2a^2 + 2a$ = $2(a^2 + a)$ $\Rightarrow$ $(a,b) = (a,a+1)$ are solutions. My motivation ...