Linked Questions

15
votes
7answers
1k views

How to find all rational solutions of $\ x^2 + 3y^2 = 7 $?

I knew that for $ x^2 + y^2 = 1$ the x and y can be expressed by introducing one more variable where $\ m=y/(x+1) $, then $\ x= 2m/(1+m^2) $ and $\ y= (1-m^2)/(1+m^2) $. What about $\ x^2 + 3y^2 = 7 $,...
10
votes
4answers
1k views

Does the equation $x^2+23y^2=2z^2$ have integer solutions?

I would like to show that the image of the norm map $\text N : \mathbb Z \left[\frac{1 + \sqrt{-23}}{2} \right] \to \mathbb Z$ does not include $2.$ I first thought that the norm map from $\mathbb Q(\...
7
votes
3answers
2k 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? ...
8
votes
4answers
537 views

$x^2+y^2+z^2=5(xy+yz+zx)$ — Is this all solutions?

Problem: Find all integers that satisfy $x^2+y^2+z^2=5(xy+yz+zx)$. Does the following parametrization give all solutions?: $x=m^2+mn-5n^2$; $y=-5m^2+9mn-3n^2$; $z=-3m^2-3mn+n^2$, where $m,n$ are ...
1
vote
3answers
494 views

Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution

Help solving over the integers: $$ax^2+by^2+cz^2+dxy+exz+fzy=0$$ where $(x_0,y_0,z_0)$ is a known integral solution and $a,b,c,d,e,f$ are integral coefficients. I found in Tito Piezas' identities the ...
5
votes
2answers
896 views

Is this a valid way to prove, that $a^2 + b^2 \neq 3c^2$ for all integers $a, b, c$ ? (except the trivial case)

Update: (because of the length of the question, I put an update at the top) I appreciate recommendations regarding the alternative proofs. However, the main emphasis of my question is about the ...
5
votes
3answers
278 views

Diophantine equation without elementary solution but with simple non elementary solution

Is there some example of an diophantine equation that satisfies: No solution is known using elementary methods. It is simple to solve using non elementary methods (e.g. using number fields). My goal ...
3
votes
2answers
690 views

Integer Solutions to an Ellipse

I'm trying to find positive integer solutions to the ellipse $$x^2 - xy + y^2 - k^2 = 0$$ where $k$ is a constant. Specifically, I already have two solutions for a given $k$, and I'm trying to find a ...
1
vote
1answer
1k views

use the lines through the point (1,1) to describe all the points on the circle $x^2 + y^2 = 2$ whose coordinates are rational numbers

a) use the lines through the point (1 1) to describe all the points on the circle $x^2 + y^2 = 2$ whose coordiates are rational numbers. b) what goes wrong if you try to apply the same procedure to ...
2
votes
4answers
510 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
0
votes
2answers
209 views

Restrictions on the [hypothetical] solutions of a ternary quadratic form

Let $r,s,t,x,y,z$ be integers, with $rst$ squarefree, such that $$rx^2+sy^2+tz^2=0.$$ I already know that by [one of] Legendre's famous theorem[s], $-rs$ must be a square modulo each prime divisor of ...
3
votes
1answer
68 views

Diophantine equation $x^2+101^2y^2=2z^2$

I tried find solutions in integers to the equation $$x^2+101^2y^2=2z^2$$ and I believe there doesn't exist one, but I keep missing how to prove it. I tried Looking square residue and similar, but ...
0
votes
3answers
117 views

Quadratic Diophantine Equation with Rational Coefficients

The problem is as below: Solve all solutions to $x^2+\dfrac{p}{q}(xy)+y^2=z^2$ for $x$, $y$, $z\in\mathbb{Q}$ and $p$, $q\in\mathbb{N}$ with $\gcd{(p,q)}=1$. My attempt: Noticing that for a ...
2
votes
1answer
117 views

How can I show $z^2 = 6x^2 + 2y^2$ has no non-trivial integer solutions?

EDIT: I missed a part of the question as there was a typo in my notes (this is part of my working on a proof using the Bruck–Ryser–Chowla theorem which was presented incorrectly in my version of the ...
0
votes
1answer
112 views

When can we solve a diophantine equation with degree $2$ in $3$ unknowns completely?

The diophantine equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ can be solved completely : for every sixtupel $(A,B,C,D,E,F)$ we can determine the complete set of integer pairs satisfying the equation. What ...

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