2
votes
1answer
85 views

Diophantine equation: $2 a^2 + 2 b^2 = c^2 + d^2$ [duplicate]

I am looking for integer solutions of the following equation: $$ 2a^{2} + 2b^{2} - c^{2} - d^{2} = 0 $$ Preferentially the solutions should obey $a+b+c+d=0$. By inspection I found the solutions: ...
1
vote
1answer
33 views

Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
1
vote
1answer
103 views

What's the approach to find out if this equation has integer solutions?

The equation is $u^3(s^4 + (r-1)^4) - s^4(t - 1)^3 = 0$ Has no integer solutions for $u,s \neq 0$. How do mathematicians today approach this problem? Sorry for broadness, just looking for a ...
3
votes
0answers
108 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
3
votes
1answer
144 views

sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$. In other words, there are finite many polynomial triples ...
2
votes
0answers
162 views

Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
4
votes
2answers
175 views

Does this equation have integer solutions

Let $g\geq 2$ be an integer. (It will be the genus of some curve.) Are there positive integers $d$ and $e$ such that the equality $$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
1
vote
2answers
344 views

Generating Pythagorean triples for $a^2+b^2=5c^2$?

Just trying to figure out a way to generate triples for $a^2+b^2=5c^2$. The wiki article shows how it is done for $a^2+b^2=c^2$ but I am not sure how to extrapolate.
1
vote
0answers
24 views

Can one determine in finite time whether a point is $S$-integral

Let $x$ be a $\mathbf{Q}$-rational point of $\mathbf{P}^1-\{0,1,\infty\}$. Let $S$ be a finite set of primes. How do I check in finite time whether $x$ is $S$-integral or not? I know how to do this ...
3
votes
1answer
118 views

Diophantine equations and Groebner bases

I'm trying to teach myself the basics of algebraic geometry and have run into something that I don't understand. I know that the problem of deciding whether a Diophantine equation $P(\vec{x}) = 0$ ...
2
votes
1answer
473 views

Parametrization of a conic and rational solutions

How can we parametrize the conic $C$: $x^2+y^2 = 5$, by considering a variable line through $(2,1)$ and hence all rational solutions of $x^2 + y^2 = 5$? I'm thinking let $x = \sqrt{5}\cos t$, and $y ...
3
votes
0answers
82 views

Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...