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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

74 questions
3 votes 0 answers 61 views

Stronger than Problem 6 from IMO 1988. [duplicate]

Let $a,b$ be positive intergers such that $ab+1\mid a^2+b^2$. Prove that $$\frac{a^2+b^2}{ab+1}=\gcd(a,b)^2$$ Using Vieta Jumping we can prove that $a^2+b^2/ab+1$ is a perfect square but I don't know ... user avatar PNT
  • 3,101
3 votes 2 answers 84 views

Understanding vieta jumping.

Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Show that $$\frac{a^2 + b^2}{ab+1}$$ is the square of an integer. I have a few questions about the proof. First here it is ,... user avatar PNT
  • 3,101
2 votes 1 answer 175 views

If $x^2 - 10ax - 11b = 0$ has roots $c$ and $d$ and $x^2 - 10cx - 11d = 0$ has roots $a$ and $b$, then find $a + b + c + d$?

If $x^2 - 10ax - 11b = 0$ have roots $c$ and $d$ and $x^2 - 10cx - 11d = 0$ have roots $a$ and $b$, then find $a + b + c + d$? My attempt- Using Vieta's formula, we get $$ c+d=10a, a+b=10c $$ Because ... user avatar user876009 9 votes 2 answers 385 views

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, ... user avatar Kyky
  • 2,625
-1 votes 1 answer 91 views

Are all the solutions produced by Mathematica?

Here is a very strong and impressive result of the Reduce command.Reduce[a^2 + b^2 == 841*(a*b + 1), {a, b}, PositiveIntegers]... user avatar user64494
  • 5,401
5 votes 4 answers 237 views

A caboodle of Pell's equation in one? $x^2+y^2-5xy+5=0$

I saw this twitter post that reads: Find all the pairs of positive integers $(x,y)$ satisfying $$ x^2 + y^2 - 5xy + 5 = 0 . $$ I don't know how to tackle this and I ended up summoning WolframAlpha ... user avatar GohP.iHan
  • 1,232
1 vote 1 answer 120 views

A Diophantine Equation Related to the Markoff Numbers

Consider the following Diophantine equation. $$5(p^2+q^2+r^2+s^2+t^2)^2-7(p^4+q^4+r^4+s^4+t^4)=90pqrst$$ This equation was discussed at The Diophantine equation $5(p^2+q^2+r^2+s^2+t^2)^2 - 7(p^4+q^4+r^... user avatar piepie
  • 518
2 votes 1 answer 123 views

Infinitely many solutions of the equation $\frac{x+1}{y}+\frac{y+1}{x} = 4$ [closed]

Prove that there exists infinitely many positive integer solutions in $(x,y)$ to the equation : $$\frac{x+1}{y} + \frac{y+1}{x} = 4$$ user avatar user636239 9 votes 4 answers 341 views

What are all possible positive integers $k$ such that $k=\frac{a^2+b^2+c^2}{bc+ca+ab}$ for some positive integers $a$, $b$, and $c$?

This question is inspired by this one. It comes in two parts. Question 1. Determine all positive integers $k$ such that there are positive integers $a$, $b$, and $c$ such that $$\frac{a^2+b^2+c^2}{... user avatar Batominovski
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2 votes 2 answers 181 views

Number Theory And Vieta Jumping [duplicate]

$\textbf{Question:}$Find all positive integers $a, b$ such that the expression $$\frac{a^2+b^2+1}{ab-1}$$ is an integer. $$$$As the expression is symmetric in $a, b$, so let $a \geq b$. It is easy to ... user avatar user786225
  • 91
5 votes 0 answers 132 views

polynomial equation $ A(x+y_1)(x+y_2)...(x+y_n) + B(x+z_1)(x+z_2)...(x+z_k) = f(x) $ ??

Consider given integers $A,B$ such that $AB \neq 0$. Consider a given polynomial $f(x) = a_0 + a_1 x + a_2 x^2 + ... $ of degree $n > 1$ with rational coefficients $a_i$. Now I wonder about ... user avatar mick
  • 12.9k
3 votes 1 answer 155 views

Find all positive integers which are representable uniquely as $\frac{x^2+y}{xy+1}$ with $x,y$ positive integers.

$\textbf{Question:}$ Find all positive integers,which are representable uniquely as $$\frac{x^2+y}{xy+1}\,,$$ where $x$ and $y$ are positive integers. I think this question maybe has something to do ... user avatar Yes it's me
  • 1,756
4 votes 1 answer 165 views

All integer values of $\frac{a^2+b^2+1}{ab-1}$

Determine all possible values of $\frac{a^2+b^2+1}{ab-1}$ where $a,b$ are positive integers. I am quite certain one should use a Vieta jumping argument, but I cannot complete it. Let $\frac{a^2+b^2+... user avatar DesmondMiles
  • 2,557
0 votes 2 answers 208 views

Find all positive integers $n$ for which the equation $x + y + u + v = n \sqrt{ xyuv }$ has a solution in positive integers. [closed]

Find all positive integers $n$ for which the equation $$ x + y + u + v = n \sqrt{ xyuv } $$ has a solution in positive integers. This problem is taken from Vietnamese Mathematical Olympiad, 2002, ... user avatar Calvin Lin
  • 60.6k
2 votes 1 answer 172 views

Find all positive integer pairs $(a, b)$ such that $(ab + a + b) \mid (a^2 + b^2 + 1)$.

Find all positive integer pairs $(a, b)$ such that $$(ab + a + b) \mid (a^2 + b^2 + 1)$$ Let $a^2 + b^2 + 1 = k(ab + a + b), k \in \mathbb N, k \ge 1$. For $k = 1$, we have that $$a^2 + b^2 - ab - a ... user avatar Lê Thành Đạt
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