All Questions
69 questions
5
votes
1
answer
236
views
Methods of finding integer solutions beyond the reach of direct search
Consider a classical problem: given a polynomial Diophantine equation $P(x_1,\dots,x_n)=0$, determine whether it has an integer solution. While this problem is undecidable in general, we may still ...
- 7,182
7
votes
2
answers
617
views
Genus 0 curves on surfaces and the abc conjecture
One of the most obvious methods to prove that a given Diophantine equation $P(x_1, \dots, x_n)=0$ has infinitely many integer solutions is to find polynomials $P_1, \dots, P_n$ in one variable $u$, ...
- 7,182
3
votes
1
answer
222
views
Large integral points on the quadratic twist $ D y^2=x^3+A x +B$
For integers $A,B,D$ and $D$ squarefree let $E : y^2=x^3+A x + B$
and $E_D$ be the quadratic twist of the elliptic curve $E$:
$$ E_D : D y^2=x^3+Ax +B$$
$E_D$ is isomorphic to $ E'_D : y^2=x^3+D^2 A ...
- 25.4k
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
14
votes
1
answer
763
views
Is 36 a sum of 4 rational fourth powers?
Hasse principle is known to hold for homogeneous quadratic equations, but fail for some 3- and 4-variable cubics, such as $5x^3+4y^3+3z^3=0$ or $15x^3+10y^3+4z^3+3t^3=0$. These counterexamples are ...
- 7,182
2
votes
2
answers
270
views
Finding rational points on intersection of quadrics in affine 3-space
Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations
\begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\
f_2 : a_2x^2 - z^2 - b_2^2 & = & 0
\end{eqnarray*}
...
- 7,527
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
18
votes
2
answers
2k
views
What is the taxicab number for rational fourth powers?
The taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in two different ways, and it is equal to $1729=12^3+1^3=10^3+9^3$. There are generalizations to ...
- 7,182
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
3
answers
606
views
Can you describe all rational solutions to these simple-looking equations?
Can you describe, in parametric form or in any other explicit way, all rational solutions to any of the following equations:
$$
y^2 + z^2 = x^3+1,
$$
$$
y^2 + z^2 = x^3-1,
$$
$$
y^2+x^2y+z^2+1=0.
$$
...
- 7,182
14
votes
1
answer
612
views
What are the rational solutions to $y^4=x^3+x+1$?
What are the rational solutions to $y^4=x^3+x+1$?
This equation is interesting because it has substitution $y^2=z$ that reduces it to elliptic curve $z^2=x^3+x+1$. Sometimes, the existence of such ...
- 7,182
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
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
$$
\quad \quad z^3 - z = x^4+y^4 \...
- 7,182
1
vote
0
answers
152
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
- 7,182
0
votes
2
answers
228
views
$y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves
What are the rational solutions to the equation
$$
y^3 = x^4 + x,
$$
in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$?
Context: This is the simplest-looking ...
- 7,182
4
votes
1
answer
916
views
Does this conic have a rational point?
Consider the conic
$$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$
over the function field $\mathbb{Q}(u,v)$.
Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
- 8,998
0
votes
1
answer
205
views
Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
- 7,182
1
vote
0
answers
98
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
2
votes
0
answers
87
views
Complexity of finding solutions of trapdoored polynomial?
Related to this question Cryptography signature scheme based on hardness of finding points on varieties.
Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.
By abuse of notation, for polynomial $f$, ...
- 25.4k
2
votes
0
answers
96
views
Cryptography signature scheme based on hardness of finding points on varieties?
Related to this question Complexity of finding solutions of trapdoored polynomial.
I am trying to build signature scheme based on hardness
of finding points on varieties.
Let $K$ be field and $M=K[x_1,...
- 25.4k
3
votes
0
answers
131
views
Rational points on a cubic surface with small coefficients
Do there exists integers $(x,y,z,t)\neq (0,0,0,0)$ such that
$$
2x^3+2y^3+z^3+t^3+2x^2y-2z^2x-y^2z-z^2t = 0 ?
$$
A short motivation: there are many known counterexamples to the Hasse principle for ...
- 7,182
7
votes
2
answers
641
views
Existence of rational points on a generalized Fermat quartic
Question: Do there exist integers $(x,y,z)\neq (0,0,0)$ such that
$$
13x^4+11y^4=8z^4 ?
$$
Some motivation: This is currently the smallest (in a sense defined here On the smallest open Diophantine ...
- 7,182
7
votes
1
answer
315
views
Rational points on regular curves over global fields
Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...
- 21.3k
4
votes
0
answers
211
views
Rational solutions to Catalan's equation
Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation
$$
x^{a}-y^{b}=1.
$$
for $a, b > 1$ and $x, y > 0$ is $x = 3,...
- 7,182
4
votes
1
answer
442
views
Trigonometric Diophantine equation
Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number?
This ...
- 1,462
3
votes
0
answers
210
views
How to find rational points on genus 2 rank 2 curves such as $y^2=x^6-4x+4$?
The question is in the title. The motivation comes from trying solving Diophantine equations in order, see Can you solve the listed smallest open Diophantine equations? . Because there is an algorithm ...
- 7,182
1
vote
2
answers
349
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
2
votes
1
answer
259
views
Rational points on a special class of surfaces
Consider a smooth surface of the following form
$$
S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3
$$
over $\mathbb{Q}$, and set
$$
U_S = \{t' \in \mathbb{...
- 8,998
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
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
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
1
answer
438
views
$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
- 7,182
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
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
4
votes
1
answer
571
views
Relation between stacky curves and "M-curves"
A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
- 1,364
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 ...
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
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
-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
6
votes
3
answers
605
views
Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$
Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$
Of course, there are solutions to this like $(a,b,c) = (9,8,6)$.
Is there any known approximation for the ...
- 749
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
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
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
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
16
votes
3
answers
1k
views
Number of solutions to polynomial congruences
Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that ...
- 3,625
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
$$
G_j(\mathbf{x}) \...
- 3,625
6
votes
0
answers
437
views
Are there infinitely many integer solutions to $a^4+b^4-c^4=N$?
Is there non-zero integer $N$ such that
$$ a^4+b^4-c^4=N \qquad (1)$$
has infinitely many integer solutions $(a,b,c)$ with $a,b \ne \pm c$?
(1) is a surface, so possible approach is to find genus 0 ...
- 25.4k
2
votes
0
answers
364
views
Find all rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$
I'm trying to classify all of the rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$.
This is a genus 2 curves and MAGMA gives a RankBound on this curve's Jacobian of ...
- 21
4
votes
1
answer
579
views
Find all rational solutions of $x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1)$
Find all rational solutions of
$$x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1).$$
Clearly the following six solutions hold:
$$(x,y)=(1,1),(-1,-1),(-1,1),(1,-1),(0,1),(0,-1)$$
But how to find all rational ...
- 4,280
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