# Questions tagged [mordell-curves]

For diophantine equations of the form $x^3=y^2+k$.

49 questions
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### Obtaining elliptic curve solution in integers from solution in quadratic field

Are there any methods or known tricks to obtain elliptic curve solutions in the integers from a solution in a quadratic field? Starting with a Mordell curve: $$y^2 = x^3 + k$$ Consider an integer $p$ ...
1answer
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### When does a Mordell curve have non-trivial torsion?

Is there a known simple criteria for when a Mordell curve has non-trivial torsion? A comment in this question: Family of elliptic curves with trivial torsion Suggests that $$y^2 = x^3 + k$$ has ...
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### Solving $y^2 = 4x^3 - p$, with prime $p \equiv 7 (\text{mod } 8)$

I'm trying to find integer solutions to equations of the form $$y^2 = 4x^3 - p \tag{1}$$ where $p$ is a prime and $p \equiv 7 (\text{mod } 8)$. 1) Is there a simple way to check if solutions do not ...
1answer
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### Solution to Mordell's Equation $y^2=x^3+4$

Here is my question: Find all solutions to $y^2=x^3+4$. My attempt: Rewrite the equation as $(y-2)(y+2)=x^3$. Notice that if $y$ is odd, then $(y-2,y+2)=1$. Hence they are both cubes, but no ...
1answer
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### Can I construct an integer giving many solutions to the Mordell-equation?

The Mordell-equation $$x^2=y^3+k$$ with integer $k\ne 0$ is known to have only finite many integer solutions $(x/y)$ Given a positive integer $n$, can I construct an integer $k$ such that there are ...
1answer
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### Finish the exercise to find or count the solutions of the equation $n x^2-\operatorname{rad}(n)=y^3$ over positive integers

I would like to solve if it is possible next diophantine equation for positive integers $n\geq 1$, $x\geq 1$ and $y\geq 1$ $$n x^2-\operatorname{rad}(n)=y^3,\tag{1}$$ where $\operatorname{rad}(n)$ ...
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### Integer solutions of $ax^3 + bx^2 + cx - y^2 = k$

The Mordell curve $$y^2=x^3+k$$ is known to have finite many solutions for every integer $k\ne 0$. How can I verify if an equation like $$ax^3 + bx^2 + cx - y^2 = k$$ with $a,b,c \in \mathbb{Z}$ have ...
2answers
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### Integer solutions for $x^3=y^2+5$

I have been able to solve the equation $x^3=y^2+a$ for integers where $1 \leq a \leq 4$ by splitting in $\Bbb Z [i\sqrt a]$. However as a natural continuation I would like to know whether the equation ...
2answers
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### Prove the Mordell's equation $x^2+41=y^3$ has no integer solutions

Why does $x^2+41=y^3$ have no integer solutions? I know how to find solutions for some of the Mordell's equations $(x^2=y^3+k)$ (using $\mathbb{Z}[\sqrt{-k}]$, and arguments of the sort). Still, I ...
0answers
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### Is it actually known that the Mordell curve has such a large solution?

Here : Examples of Diophantine equations with a large finite number of solutions if I understood it right, in one of the answers it is claimed that the Mordell-equation $$y^2=x^3+154319269$$ has a ...
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### Is there an easy criterion whether a number is the sum of a square and a cube?

The solutions of the Mordell curve $y^2=x^3+k$ with NON-POSITIVE $x$-value can be efficiently found by brute force. This can be reformulated as follows : $k$ is the sum of a square and a (non-negative)...
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### Has the Mordell-curve equation always a solution in $\mathbb Z_n$?

I searched numbers $n$ and $k$, such that the equation $$y^2=x^3+k$$ (Modell-curve-equation) has no solution modulo $n$ in order to find families of $k$'s such that $y^2=x^3+k$ has no integral ...
1answer
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### How large can the smallest integral solution of a Mordell curve be?

Suppose, a Mordell curve $$y^2=x^3+k$$ has at least one integral solution. Denote $d$ to be the smallest absolute value of $x$, such that $x^3+k$ is a square, in other words, $d$ is the the smallest ...
1answer
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### For which small $n$ is unknown whether the Mordell-curve has an integral point?

Since it is not easy to determine the integral points of a Mordell curve $$y^2=x^3+n$$ with integer $n\ne 0$, I came to the following questions : $1)$ What is the smallest (in absolute value) ...
1answer
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### Is there an explanation for these gaps?

Here : http://www.math.ubc.ca/~bennett/BeGa-data.html there is a complete list of the integral points of the Mordell-curves for $-10^7\le n\le 10^7$. I searched for large solutions (in particular ...
1answer
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### Which Mordell-curves have exactly $2$ integral points with large coordinates?

I found the list of the integral solutions of the Mordell-equation $$y^2=x^3+n$$ with non-zero integer $n$ for $-10^7\le n\le 10^7$ I am interested in integers $n$ such that the Mordell-equation has ...
1answer
164 views

### How to find an upper bound on the number of solutions of $y^3=x^2+4^k$

I have solved the first two parts of this question but I am struggling with the remaining section. I can't see any meaningful way to reuse what I did before and/or find a way forward. Just to be ...
1answer
242 views

### Mordell curves with many integral points

For $k\in{\mathbb Z},k\neq 0$, denote by $f(k)$ the number of integral points on the Mordell curve $y^2-x^3=k$. According to the data at http://tnt.math.se.tmu.ac.jp/simath/MORDELL , the largest value ...
2answers
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### Are all Mordell equations $y^2=x^3+k$, for any integer $k$, solvable

Are all Mordell equations $y^2=x^3+k$, for any integer $k$, solvable? Not that there are solutions $x,y$ for every $k$, but that you can determine for every $k$ if there are solutions, and if there ...
2answers
238 views

### Is $y^2=x^3+7$ unsolvable modulo some $n$?

The equation $y^2=4x^3+7$ has no integral solution since $y^2\equiv4x^3+7\pmod4$ has no solution (i.e. has no solution in $\Bbb{Z}/4\Bbb{Z}$). It is well known that $y^2=x^3+7$ has no integral ...
1answer
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### find all integer solutions of $y^2=x^3-2$ [duplicate]

I’m blind about integer solutions of a polynomial. I have no number theory background, but I’m curious about how to figure out all integer solutions of a polynomial, for example this question. It is ...
0answers
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### Elementary solution to the Mordell equation $y^2=x^3+9$?

I've recently been wondering how to solve the equation of mordell for $k=9$, namely: $y^{2}=x^{3}+9.$ It reduced to solving the Thue equation $\lvert\,a^{2}-2b^{3}\rvert=3.$ Interestingly, the ...
2answers
211 views

### How can we solve $y^2=x^3+23$ without trial and error? [duplicate]

$$y^2=x^3+23$$ Are there any easy ways to solve this problem with number theory, abstract algebra, etc.? (trial and error for mods by the way)
1answer
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### $x^3-9=y^2$ find integral solutions

Find all integral solutions $x^3-9=y^2$ I tried many times but still no idea how to solve it. I will be grateful for any help.
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1answer
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### How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
3answers
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### How could I calculate the rank of the elliptic curve $y^2 = x^3 - 432$?

The birational change of variables $(u,v) = (\frac{36+y}{6x},\frac{36-y}{6x})$ maps $u^3+v^3=1$ to $y^2 = x^3 - 432$ which has discriminant $-2^{12}\cdot 3^9$. Using pari/gp we can compute the ...