# Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

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### Diophantine Simultaneous Equations

Solve the set of Diophantine equations: $\frac{x}{y} - \frac{1}{y^2} = a^2 - \frac{64}{x}$ $a^{2x} + 9 = y - \frac {1} {x^2}$ Edit: I have got as far as this (by multiplying, simplifying and solving ...
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### Given $n\geq 0, j\geq 1.$ How many solutions to: $\sum_{i=1}^j x_i=n,$ where $x_i\geq 0$ and $x_i$ are all different?

Given $n\in\mathbb{N}\cup\{0\},\ j\in\mathbb{N}.$ How many solutions are there to: $\sum_{i=1}^j x_i=n,$ where $x_i\in\mathbb{N}\cup\{0\}$ for all $i\in\{1,\ldots,j\}$ and $x_k\neq x_l$ if $k\neq l\ ?$...
1 vote
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### Find all pairs $(a,b)$ of positive integers satisfying $6a^2+a=b^2$.

I have already tried treating the equation as a quadratic on a and b, but it doesn't work. I also have plugged in some values. $(6,10)$ is a solution, but I didn't manage to find any other. Are there ...
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### A Diophantine equation along with an inequality

Find all positive integers $a, b, c$ and integers $x, y, z$ such that \begin{align*} ax^2+by^2+cz^2=abc+2xyz-1\\ ab+bc+ca\geq x^2+y^2+z^2 \end{align*} My Progress - By $\pmod 4$. I could conclude ...
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### Rational points on $y^2=5(x^4+1)$

How would one find $\mathbb{Q}$ -rational points on $y^2=5(x^4+1)$. I do not think there is one but I would also love to see a proof of it. I thought about writing them in terms of fractions and got ...
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### Solve $4(x+y+z+4) = xyz$ [closed]

How do I solve the following equation: $xyz = 4(x+y+z+4)$, subject to the conditions that $x,y,z$ are positive integers?
1 vote
### Solution to Pell's equation $a^2 - Nb^2 = 1$ when $a$ is an even integer
Given an integer $a$, show that there is a square-free integer $N > 1$ and an integer $b$ such that $a^2 - Nb^2 = 1$. I'm not too sure how to go about proving this. I believe I'm supposed to use ...