0
votes
0answers
40 views

Finding all integral solutions of a positive definite quadratic equation

Let $q(x_1,\ldots,x_n)$ be an integral positive definite quadratic form. For $d\in\mathbb{N}$ the equation $$q(x_1,\ldots,x_n)=d$$ has a finite number of integral solutions. Is there an algorithm to ...
4
votes
5answers
152 views

Looking for proof of no solution to 4-variable quadratic diophantine equation

Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$ Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the ...
2
votes
1answer
71 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

$a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
0
votes
1answer
60 views

What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?

Actually, I need to find if $4x^2 - 3y^2 - z^2 = 12$ is solvable. But I somehow feel that applying theory of integer representation by quadratic forms in three variables would yield quicker results... ...
0
votes
1answer
55 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
0
votes
0answers
67 views

How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$?

I am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!
2
votes
1answer
60 views

finding zeroes of a quadratic form

Let $a,b\in\mathbb Z$ be squarefree with $a>0$. Suppose that I know that there exist $(0,0,0)\neq (x,y,z) \in \mathbb Z^3$ s.t. $x^2-by^2-az^2=0$. Is there any known algorithm to find any such a ...
3
votes
6answers
298 views

Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
7
votes
2answers
827 views

Show $15x^{2} - 7y^{2} = 9$ has no integer solutions

I'm trying to show the quadratic binary has no integer solution. I've used the following process to transform it into a Pell's equation of the form $x^{2} - Dy^{2} = M$ If there is a solution, then ...
1
vote
2answers
336 views

Finding positive integer solutions to $n = ax^2 +by^2 - cxy$

How can I find the positive integer solutions to $x$ and $y$, given that $n$, $a$, $b$ and $c$ are all positive integers, in an equation of the form: $$n = ax^2 + by^2 - cxy.$$ Specifically, I want ...
0
votes
3answers
165 views

Numbers of the form $x^2+axy+by^2$

This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation $x^2+axy+by^2=u^2+auv+bv^2$ The title of the ...
1
vote
1answer
299 views

Find all solutions of this diophantine equation of the second degree in three variables

Consider the Diophantine equation $Q(x,y,z)=1$, where $Q(x,y,z)$ is the quadratic form $x^2+y^2-z^2$. Let $S \subseteq {\mathbb Z}^3$ denote the set of all solutions. It is rather easy to find several ...
2
votes
2answers
186 views

A question about integral quadratic forms

Hi Would you please advise me? Consider the equation below: $$ ax^2+bxy+cy^2=n $$ in which $a, b, c$ and $n$ are integers. We then suppose that $a, b, c$ are constant. Is there any way to find the ...
2
votes
3answers
615 views

Existence of solutions to diophantine quadratic form

Is there a general result about the existence of (non-trivial) solutions of the diophantine equation: $$Ax^2 + By^2 = Cz^2$$ for A,B,C known positive integers, pair-wise relatively prime? What if ...
1
vote
3answers
720 views

How to solve inhomogeneous quadratic forms in integers?

If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ ...