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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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 ... elementary-number-theory contest-math vieta-jumping 
PNT
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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 ,... proof-explanation contest-math vieta-jumping PNT
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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 ... polynomials roots vieta-jumping 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, ... number-theory diophantine-equations math-history vieta-jumping Kyky
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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] ... diophantine-equations vieta-jumping user64494
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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 ... diophantine-equations quadratics recursion pell-type-equations vieta-jumping GohP.iHan
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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^... number-theory diophantine-equations vieta-jumping piepie
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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 : $$\frac{x+1}{y} + \frac{y+1}{x} = 4$$ elementary-number-theory diophantine-equations fractions pell-type-equations vieta-jumping 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 $$\frac{a^2+b^2+c^2}{... number-theory elementary-number-theory diophantine-equations quadratic-forms vieta-jumping Batominovski
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Number Theory And Vieta Jumping [duplicate]
$\textbf{Question:}$Find all positive integers $a, b$ such that the expression $$\frac{a^2+b^2+1}{ab-1}$$ is an integer. $$$$As the expression is symmetric in $a, b$, so let $a \geq b$. It is easy to ... number-theory elementary-number-theory contest-math solution-verification vieta-jumping user786225
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polynomial equation $ A(x+y_1)(x+y_2)...(x+y_n) + B(x+z_1)(x+z_2)...(x+z_k) = f(x) $ ??
Consider given integers $A,B$ such that $AB \neq 0$. Consider a given polynomial $f(x) = a_0 + a_1 x + a_2 x^2 + ... $ of degree $n > 1$ with rational coefficients $a_i$. Now I wonder about ... number-theory polynomials diophantine-equations factoring vieta-jumping mick
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Find all positive integers which are representable uniquely as $\frac{x^2+y}{xy+1}$ with $x,y$ positive integers.
$\textbf{Question:}$ Find all positive integers,which are representable uniquely as $$\frac{x^2+y}{xy+1}\,,$$ where $x$ and $y$ are positive integers. I think this question maybe has something to do ... elementary-number-theory contest-math divisibility diophantine-equations vieta-jumping Yes it's me
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All integer values of $\frac{a^2+b^2+1}{ab-1}$
Determine all possible values of $\frac{a^2+b^2+1}{ab-1}$ where $a,b$ are positive integers. I am quite certain one should use a Vieta jumping argument, but I cannot complete it. Let $\frac{a^2+b^2+... elementary-number-theory vieta-jumping DesmondMiles
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Find all positive integers $n$ for which the equation $x + y + u + v = n \sqrt{ xyuv }$ has a solution in positive integers. [closed]
Find all positive integers $n$ for which the equation $$ x + y + u + v = n \sqrt{ xyuv } $$ has a solution in positive integers. This problem is taken from Vietnamese Mathematical Olympiad, 2002, ... elementary-number-theory contest-math vieta-jumping Calvin Lin
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Find all positive integer pairs $(a, b)$ such that $(ab + a + b) \mid (a^2 + b^2 + 1)$.
Find all positive integer pairs $(a, b)$ such that $$(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 ... number-theory vieta-jumping Lê Thành Đạt
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