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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 ...
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
41
votes
Accepted
Rational points on the "quintic circle" $x^5 + y^5 = 7$
There is an action of $\mu_5$, the group of fifth roots of unity, on your curve,
given by $\zeta \cdot (x,y) = (\zeta x, \zeta^{-1} y)$. The quotient by this
group action is the hyperelliptic curve
$$ …
- 11.3k
35
votes
Accepted
Is equation $xy(x+y)=7z^2+1$ solvable in integers?
There is no solution.
It is clear that at least one of $x$ and $y$ is positive and that neither is divisible by 7. We can assume that $a := x > 0$. The equation implies that there are integers $X$, $Y …
- 11.3k
22
votes
Fermat's last theorem over larger fields
This is not quite an answer, but not quite a comment either.
We can at least show that $X(({\mathbb Q}^{\text{ab}})^{\text{ab}})$ is infinite (where $X$ is the quintic Fermat curve). There are in fac …
- 11.3k
18
votes
Accepted
rational points of a hyperelliptic curve
By now there is a fairly rich literature on computing the set of rational points on curves of higher genus, see for example my survey paper on "Rational points on curves".
What one can do for your co …
- 11.3k
18
votes
Accepted
Special arithmetic progressions involving perfect squares
Starting from the equations in my previous answer, we get, by multiplying them in pairs,
$$(x-y)x(x+y)(x+2y) + (x-y)x + (x+y)(x+2y) + 1 = (z_1 z_6)^2\,,$$
$$(x-y)x(x+y)(x+2y) + (x-y)(x+y) + x(x+2y) + …
- 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
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
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
14
votes
what is the maximum number of rational points of a curve of genus 2 over the rationals
Here is some more information.
The curve that establishes the current record is obtained from a K3
surface $S$ that was found by Noam Elkies. $S$ is a double cover of
${\mathbb P}^2$ ramified above a …
- 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
12
votes
Find all rational solutions of this diophantine-equation?
Taking the equation in Joe Silverman's comment as the defining equation and asking Magma:
> A := AffineSpace(Rationals(), 2);
> C := Curve(A, q^2*p^4 + (-4*q^3+4*q)*p^3 - 2*q^2*p^2 + (4*q^3-4 …
- 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
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
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