Tagged Questions
6
votes
2answers
136 views
Solve: $x^2-py^2=q$
Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers.
It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol.
...
1
vote
3answers
64 views
Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$
For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$
where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
4
votes
1answer
46 views
On Selmer's curve
I am trying to prove that the equation $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ has non-trivial solutions for all primes $p$. I divide it into 3 cases: $p \equiv 0,1,2 \pmod{3}$. The cases $p \equiv 0,2 ...
1
vote
0answers
73 views
Solve the equation $x^4+y^4=d*z^2$
Solve the equation:$$x^4+y^4=d*z^2,$$
where $x,y,z$ are positive integers,and $d>1$ is a given square-free integer.
I know if $p$ is an odd prime and $p|d,$ then $t^4\equiv -1 \pmod p$ is ...
1
vote
2answers
94 views
How to prove that there exist infinitely many integer solutions to the equation $x^2-ny^2=1$ without Algebraic Number Theory
I learned in my Intro Algebraic Number Theory class that there exist infinitely many integer pairs $(x,y)$ that satisfy the hyperbola $x^2-ny^2=1$; just consider that there are infinitely many units ...
2
votes
1answer
101 views
If $x^p +y^p = z^p$ and $xyz \neq 0$, then $p$ divides $x$ or $y$ or $z$?
I am working on an exercise: If $x^5 +y^5 = z^5$ and $xyz \neq 0$, then $5$ divides at least one of $x$, $y$ or $z$.
I am thinking that the answer involves an application of Kummer's theorem, but I'm ...
4
votes
1answer
132 views
Solving $x^2+19=y^5$
I was given several exercises and there is a particular one, I am not able to solve.
Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
5
votes
1answer
74 views
Quadratic Diophantic equation
Hello :) i want to give a answer op the following question:
For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$.
I want to solve the question without ...
4
votes
2answers
137 views
Does this equation have integer solutions
Let $g\geq 2$ be an integer. (It will be the genus of some curve.)
Are there positive integers $d$ and $e$ such that the equality
$$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
1
vote
0answers
23 views
Can one determine in finite time whether a point is $S$-integral
Let $x$ be a $\mathbf{Q}$-rational point of $\mathbf{P}^1-\{0,1,\infty\}$.
Let $S$ be a finite set of primes. How do I check in finite time whether $x$ is $S$-integral or not?
I know how to do this ...
2
votes
2answers
91 views
Does the equation have infinite number of solutions in integers?
Does either of the equations ${a^2} - 10{b^2} = \pm 1$ have infinite number of solutions in integers?
If the answers is yes, a hint about how to reduce this problem to the problem of Pythagorean ...
4
votes
1answer
207 views
For which $(n,m,l)$ does $L_n^2+L_m^2=L_l^2$ hold?
Let $L_n = 2 \sin(\frac{\pi}{n})$ be the length of a side of the regular $n$-gon inscribed in a circle.
For which $(n,m,l)$ does $$L_n^2+L_m^2=L_l^2$$ hold?
I found by computer search that (3,6,2), ...
1
vote
1answer
58 views
Clarifying a step in the solution of $y^2 + 2 = x^3$
In Ian Stewart's book on Algebraic Number Theory, he gives the following proof that the solutions to $y^2 + 2 = x^3$ is $x=3, y = \pm 5$:
I'm confused about the step "since $c^2 + 2d^2 | 4y^2$ and ...
3
votes
1answer
197 views
Diophantine equation (use class ideal group to solve)
Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$
My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
4
votes
1answer
180 views
Solving a Diophantine Equation using factorisation of ideals
I am stuck on the following question which is given as follows:
Prove that the only integer solutions to the equation
\begin{equation}
x^2 + 13 = y^3
\end{equation}
are $(70,17)$ and $(-70, 17)$.
...
3
votes
0answers
74 views
Do the solutions to the unit equation lie dense in the complex numbers
Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers.
...
5
votes
3answers
711 views
Integral solutions to $y^{2}=x^{3}-1$
How to prove that the only integral solutions to the equation $$y^{2}=x^{3}-1$$ is $x=1, y=0$.
I rewrote the equation as $y^{2}+1=x^{3}$ and then we can factorize $y^{2}+1$ as $$y^{2}+1 = (y+i) \cdot ...
