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Results tagged with diophantine-equations
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user 21146
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
4
votes
Algorithm for computing rational points if the rank of Jacobian is 0
Here is a sketch of a method that could be able to do what you want. (I won't call it an algorithm, since I cannot prove that it will always work.) I will write $X$ for the curve and $J$ for its Jacob …
- 11.3k
6
votes
Accepted
Algorithm for computing rational points if the rank of Jacobian is 0
There is an algorithm due to Bjorn Poonen (Computing torsion points on curves, Experiment. Math. 10 (2001), no.3, 449–465) that, given a (not necessarily rational) base-point $P_0$ on the curve, finds …
- 11.3k
7
votes
Rational points on genus 3 curves defined by short equations
I have looked a bit at the first equation. It has (at least) seven rational points (as a projective curve). The differences of these points generate a free abelian group of rank three in the Mordell-W …
- 11.3k
11
votes
Accepted
Existence of rational points on generalized Fermat quintics
Both curves have no rational points.
Curves of this shape map to genus 2 curves of the form $y^2 = a x^5 + b$
(one can make $a = 1$ if one likes), by quotienting out by the group
of automorphisms gene …
- 11.3k
35
votes
Accepted
Is equation $xy(x+y)=7z^2+1$ solvable in integers?
There is no solution.
It is clear that at least one of $x$ and $y$ is positive and that neither is divisible by 7. We can assume that $a := x > 0$. The equation implies that there are integers $X$, $Y …
- 11.3k
10
votes
Accepted
On the Diophantine equation $x^{5} + y^5 = z^p$
To the best of my knowledge, this is open for general $p$.
As mentioned by Alapan Das, Bjorn Poonen has solved the case
$p = 2$ and also $p = 3$ [B. Poonen, Some diophantine equations
of the form $x^n …
- 11.3k
7
votes
Are all partial consecutive harmonic subsums distinct?
Here is a partial result:
If we fix $a < b$, then there are at most finitely many $(c,d)$ such that
$H(a,b) = H(c,d)$.
First note that, from the asymptotics, $d-c$ must get large with $c$.
Now consi …
- 11.3k
17
votes
Accepted
Find all rational solutions of $x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1)$
This is no complete answer yet, but may get expanded to one in due course.
First we search for points on the curve, which is isomorphic to
$$C \colon y^2 = 2(x^6 - x^2 + 2);$$
this produces points wi …
- 11.3k
12
votes
What is the rank of the Mordell equation $y^2 = x^3 - 2$?
You could have a look at this paper:
M. Stoll, On the arithmetic of the curves $y^2 = x^\ell + A$, II;
J. Number Theory 93, 183-206 (2002).
Corollary 2.1 says that for $A = -2$, one gets a rank bo …
- 11.3k
17
votes
Are there any rational solutions to this octic equation?
Considering your equation as a quadratic equation in $m$, it is equivalent
to $$y^2 = 4 x^9 + 1$$ (with $y = 2xm + 1$). A solution will in particular
give a rational point on the elliptic curve $E \co …
- 11.3k
14
votes
Accepted
Imprimitive solutions to $x^2+y^3=z^7$
You can find the solutions for any given $z$ by looking for the integral
points on the elliptic curve
$$x^2 = (-y)^3 + z^7$$
(which would usually be written $y^2 = x^3 + z^7$). The curve
is isomorphic …
- 11.3k
4
votes
Accepted
On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk
The possible $m_k$, when assumed to be squarefree, must be divisors of the resultant of $x^2 + a$ and $x^2 + b$, which is $(a-b)^2$; so $m_k \mid a-b$. (Note that bot factors must be in the same squar …
- 11.3k
132
votes
Accepted
Estimating the size of solutions of a diophantine equation
This problem turned out to be much more interesting than I originally
thought. Let me give my solution, which seems to be slightly different from
(but essentially the same as) the solution in the pape …
- 11.3k
10
votes
Accepted
Why are some solutions of these diophantine equations off the usual patterns?
I take this from my comments above and add something.
The question is about rational points on the surface $S$ given by
$$ \Delta(a,b,c) := (ab+bc+ca)^3 - 27(abc)^2 = z^2 $$
in the weighted projectiv …
- 11.3k
9
votes
rational numbers and triangular numbers
There should always be solutions unless $kn$ is a square. The equation is
equivalent to
$$k (2a+1)^2 - n (2b+1)^2 = k - n.$$
Let $(x_0, y_0)$ be
the fundamental solution of the Pell equation $x^2 - 4 …
- 11.3k