Tagged Questions
3
votes
1answer
92 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
95 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
137 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
175 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
23 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
74 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
185 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
74 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.
...
