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

# Questions tagged [mordell-curves]

57 questions

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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 torsion subgroup:
? elltors(ellinit([0,0,0,0,-432]))
%1…

anon

**12**

votes

**10**answers

### Solve $x^2+2=y^3$ using infinite descent?

Just so this doesn't get deleted, I want to make it clear that I already know how to solve this using the UFD $\mathbb{Z}[\sqrt{-2}]$, and am in search for the infinite descent proof that Fermat claimed to have found.
I've alaways been fascinated by…

Is Ne

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**10**

votes

**1**answer

### Proof that $y^2=x^3+21$ has no integral solution with elementary methods?

I tried to prove that $$y^2=x^3+21$$ has no integral solution in the way as shown here : http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf
My work so far : $x$ cannot be even because we would have $y^2\equiv 5\mod 8$ , which is a…

Peter

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### What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation
$$Y^2 = X^3-k \tag{1}$$
where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, though there are various ad hoc methods which…

Kieren MacMillan

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**8**

votes

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### 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 solution, but is there an integer $n (n>1)$ such that …

sacch

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### Solutions to $y^2 = x^3 + k$?

The equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 divids m, doesn't have any answer and its proof can be obtained by using quadratic reciprocity law.
Do you know…

BenyaminH

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### Finding solutions to $y^2 = x^3 - 27$

I am trying to find integer solutions to this equation:
$$ y^2 = x^3 - 27 $$
With the other problem I tried I was able to use unique factorization in $\mathbb Z [\sqrt{n}]$. I don't know how to get started on this one. I tried looking online and…

cgiusepe

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**6**

votes

**1**answer

### Mordell equation with prime-squared constant

I'm interested in a specific case of the Mordell equation:
$$E: y^2=x^3+k$$
where $k=p^2$ for some prime $p$.
Most of the literature I've been able to find regarding the Mordell equation either explicitly assumes $k$ to be square-free or avoids the…

John Gilling

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**2**answers

### I think I found an error in a OEIS-sequence. What is the proper site to post it?

I checked the link given to this OEIS-sequence :
https://oeis.org/A081121
and apparantly the numbers $3136$ and $6789$ appear in the sequence. However, we have $$4192^2=260^3-3136$$ and $$94^2=25^3-6789$$ so the two numbers should not appear in the…

Peter

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**1**answer

### When is $c^4-72b^2c^2+320b^3c-432b^4$ a positive square?

In trying to solve a certain [third-degree] Diophantine equation, I have used the quadratic equation to determine that
$$c^4-72b^2c^2+320b^3c-432b^4$$
must be a positive integer square, where $c$ and $b$ are positive integers.
From [admittedly…

Kieren MacMillan

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**6**

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**2**answers

### Integral solutions to $56u^2 + 12 u + 1 = w^3$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$
This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$? and is likely to be easier.
Note after getting…

Will Jagy

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**5**

votes

**1**answer

### 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 under some certain conditions, one can directly…

awllower

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### Solutions to the Mordell Equation modulo $p$

It is well known that for any nonzero integer $k$ the Mordell Equation $x^2 = y^3 + k$ has finitely many solutions $x$ and $y$ in $\mathbf Z$, but it has solutions modulo $n$ for all $n$. One proof of this involves using the Weil Bound to show that…

Mayank Pandey

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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 exist for a given $p$?
2) Is there a…

JustinBalenti

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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 can't find a way to prove this particular equation…

Laura

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