All Questions
30 questions with no upvoted or accepted answers
9
votes
0
answers
274
views
$y^3=x^4+x+1$, and rational points on rank 2 Picard curves
What are (a) integer, (b) rational solutions to the equation
$$
y^3 = x^4 + x + 1 ?
$$
There are obvious solutions $(x,y)=(-1,1)$ and $(0,1)$, are they the only ones?
Context: There are a lot of ...
- 7,182
6
votes
0
answers
437
views
Are there infinitely many integer solutions to $a^4+b^4-c^4=N$?
Is there non-zero integer $N$ such that
$$ a^4+b^4-c^4=N \qquad (1)$$
has infinitely many integer solutions $(a,b,c)$ with $a,b \ne \pm c$?
(1) is a surface, so possible approach is to find genus 0 ...
- 25.4k
5
votes
0
answers
454
views
Is 136 a difference of two rational fourth powers?
There is a rich literature that studies which small positive integers are the sums of two rational fourth powers, see e.g. Section 6.6 of Henri Cohen's book Volume I: Tools and Diophantine Equations. ...
- 7,182
5
votes
0
answers
215
views
Integer points of rational function in 2 variables
Is there an algorithm that, given polynomials $P(x)$ and $Q(y)$ with integer coefficients, decides whether there exists integers $x$ and $y$ such that $\frac{P(x)}{Q(y)}$ is an integer?
This is a ...
- 7,182
5
votes
0
answers
308
views
Algorithm for solutions to quadratic forms over number fields
Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)?
I am especially interested in the quaternary case. There exist some ...
- 101
4
votes
0
answers
307
views
Equations involving sum of fourth powers
Do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - 1 = x^4+y^4 \neq 0 \tag{$a$} \quad ?
$$
Also, do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - z = x^4+y^4 \...
- 7,182
4
votes
0
answers
211
views
Rational solutions to Catalan's equation
Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation
$$
x^{a}-y^{b}=1.
$$
for $a, b > 1$ and $x, y > 0$ is $x = 3,...
- 7,182
4
votes
0
answers
331
views
Counting Special Rational Points on Cubic Surfaces
A paper of Heath-Brown gives an heuristic argument for the density of rational points on two cubic surfaces: $x^3+y^3+z^3=kw^3,k=2,3$, say, the number of rational points of height less than $N$ on ...
- 3,337
3
votes
0
answers
131
views
Rational points on a cubic surface with small coefficients
Do there exists integers $(x,y,z,t)\neq (0,0,0,0)$ such that
$$
2x^3+2y^3+z^3+t^3+2x^2y-2z^2x-y^2z-z^2t = 0 ?
$$
A short motivation: there are many known counterexamples to the Hasse principle for ...
- 7,182
3
votes
0
answers
210
views
How to find rational points on genus 2 rank 2 curves such as $y^2=x^6-4x+4$?
The question is in the title. The motivation comes from trying solving Diophantine equations in order, see Can you solve the listed smallest open Diophantine equations? . Because there is an algorithm ...
- 7,182
3
votes
0
answers
84
views
Low height integer points on a rank variety
Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition
$$
\mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1.
...
- 960
3
votes
0
answers
122
views
Curves on hypersurfaces generated by diagonal sums
This is related to an earlier question of mine ((Non-)Existence of curves of low degree on affine and projective varieties). It seems that the question is too difficult for specific surfaces, although ...
- 26.9k
2
votes
0
answers
52
views
Infinitely many coprime solutions of $F(x,y)= k(a_1 x + a_2 y)^2 z^2$?
This might be related to an open problem.
Let $F(x,y)$ be homogeneous degree 4 squarefree polynomial
with integer coefficients and
$h(x,y)=a_1 x + a_2 y$ and $\gcd(F,h)=1$ and $k$ be integer.
Consider ...
- 25.4k
2
votes
0
answers
184
views
Will Coppersmith's method work for this bivariate modular polynomial shape?
I have a bivariate modular polynomial of shape
$$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$
where
$q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$,
$g(x)\in\mathbb Z[x]$ is of degree four and
$f(...
- 13.9k
2
votes
0
answers
87
views
Complexity of finding solutions of trapdoored polynomial?
Related to this question Cryptography signature scheme based on hardness of finding points on varieties.
Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.
By abuse of notation, for polynomial $f$, ...
- 25.4k
2
votes
0
answers
96
views
Cryptography signature scheme based on hardness of finding points on varieties?
Related to this question Complexity of finding solutions of trapdoored polynomial.
I am trying to build signature scheme based on hardness
of finding points on varieties.
Let $K$ be field and $M=K[x_1,...
- 25.4k
2
votes
0
answers
242
views
Solving $x^k+y^k+z^k=w^k$ non-trivially in strictly positive integers
Consider the equation $x^k+y^k+z^k=w^k$ in $x$, $y$, $z$ and $w$ with $k\in\mathbb{N}_{\geq2}$.
If we look for solutions that are strictly positive and non-trivial i.e. $x\neq-y$, $x\neq w$ etc... ...
- 4,862
2
votes
0
answers
147
views
Genus Zero Diophantine Equations and Infinite Valuations
I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:
[1] Solving genus zero diophantine ...
- 211
2
votes
0
answers
103
views
Bound for the number of solutions to a system of congruence relations
Suppose I have $n$ polynomials in $n$ variables $G_j(x_1, \ldots, x_n)$ with integer coefficients. Let $u_j$ be some fixed $p$-adic integers.
Consider the system of congruences
$$
G_j(\mathbf{x}) \...
- 3,625
2
votes
0
answers
364
views
Find all rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$
I'm trying to classify all of the rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$.
This is a genus 2 curves and MAGMA gives a RankBound on this curve's Jacobian of ...
- 21
2
votes
0
answers
242
views
Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$
when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$
I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that
$(x_{i},y_{i}),i=1,2,\cdots,N$ ...
- 4,280
1
vote
0
answers
152
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(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}$$
where ...
1
vote
0
answers
188
views
How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?
Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial
solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$.
Q1 How small $u_k$ can be infinitely often as function $k$?
This ...
- 25.4k
1
vote
0
answers
199
views
Class number of the cyclotomic tower
Let ${\Bbb Q}(\zeta_{\infty})$ be the field obtained by adjoining all roots of unity. We define
Cl(${\Bbb Q}(\zeta_{\infty})$)$\colon= \underset{m > 1}{\varinjlim}~{\mathrm{Cl}}({\Bbb Z}[\zeta_m])...
- 781
1
vote
0
answers
275
views
Integral points on affine rational curves over $\mathbb{Q}$
Given a rational curve $C:(f_1(t),f_2(t))$, where $f_i(t),i=1,2$ are rational functions with rational coefficients.
Question: Is there any criterion(proved or conjectural) for the existence of ...
- 3,337
0
votes
0
answers
110
views
Common integer roots of polynomials
I have two polynomials of form
$$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