All Questions
69 questions
2
votes
0
answers
364
views
Find all rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$
I'm trying to classify all of the rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$.
This is a genus 2 curves and MAGMA gives a RankBound on this curve's Jacobian of ...
- 21
2
votes
0
answers
242
views
Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$
when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$
I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that
$(x_{i},y_{i}),i=1,2,\cdots,N$ ...
- 4,280
1
vote
2
answers
643
views
Describe all integer/rational solutions to $x^3+y^3+z^3+t^3+s^3=0$
The question is in the title.
Equation $\sum_{i=1}^n x_i^3 = 0$ has no non-trivial integer solutions for $n=3$. For $n=4$, there are known descriptions of all integer/rational solutions, see
Choudhry, ...
- 7,182
1
vote
2
answers
349
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
1
vote
1
answer
180
views
On integral points of $f(x,y)=z g(x,y)$
Let $f(x,y),g(x,y)$ be polynomials with integer coefficients.
Consider the surface
$$ f(x,y)=z g(x,y) \qquad (1)$$
(1) has parametrization over the rationals given by
$z=\frac{f(x,y)}{g(x,y)}$.
Q1 ...
- 25.4k
1
vote
0
answers
152
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
- 7,182
1
vote
0
answers
98
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
1
vote
0
answers
261
views
Integer points on genus 1 curves using CAS
How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.?
As a specific example, do ...
- 7,182
1
vote
0
answers
192
views
Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $
I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$.
In other words,
$$y^2=1+4n^{\underline k},\tag{I}$$
where ...
1
vote
0
answers
188
views
How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?
Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial
solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$.
Q1 How small $u_k$ can be infinitely often as function $k$?
This ...
- 25.4k
1
vote
0
answers
199
views
Class number of the cyclotomic tower
Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define
Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...
- 781
1
vote
0
answers
275
views
Integral points on affine rational curves over $\mathbb{Q}$
Given a rational curve $C:(f_1(t),f_2(t))$, where $f_i(t),i=1,2$ are rational functions with rational coefficients.
Question: Is there any criterion(proved or conjectural) for the existence of ...
- 3,337
0
votes
1
answer
325
views
On the elliptic curve $y^2 = x^3 + z^{4k}$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some $k \in \mathbb{Z}_{>1}$ ?
- 1,019
0
votes
2
answers
228
views
$y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves
What are the rational solutions to the equation
$$
y^3 = x^4 + x,
$$
in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$?
Context: This is the simplest-looking ...
- 7,182
0
votes
1
answer
285
views
The number of solutions of a Diophantine equation [closed]
Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...
- 11.3k
0
votes
1
answer
204
views
Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
- 7,182
0
votes
0
answers
110
views
Common integer roots of polynomials
I have two polynomials of form
$$f_1(w,x)=M_1$$
$$f_2(y,z)=M_2$$
and I have two polynomials of form
$$g_1(w,x,y,z)=M_3$$
$$g_2(w,x,y,z)=M_4$$
where $f_1,f_2,g_1,g_2\in\mathbb Z[w,x,y,z]$ and $M_1,M_2,...
- 13.9k
0
votes
0
answers
96
views
Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...
- 25.4k
-3
votes
2
answers
608
views
Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
- 1,019