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Questions tagged [mordell-curves]

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

3
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1answer
64 views

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 $p$ ...
1
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1answer
68 views

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 ...
5
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2answers
96 views

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 ...
1
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1answer
120 views

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 ...
2
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1answer
53 views

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 ...
1
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1answer
56 views

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 $\operatorname{rad}(n)$ ...
2
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0answers
73 views

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 ...
3
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2answers
97 views

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 ...
4
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2answers
440 views

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 ...
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0answers
118 views

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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0answers
73 views

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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0answers
39 views

Can we check efficiently whether a given integer solution of a Mordell-curve is the smallest?

Suppose, we know an integer solution $(y,x)$ of a Mordell-curve $$y^2=x^3+k$$ For example $$25124268633183975113^2=8578189162349^3-251669431780$$ So for $$k=-251 669 431 780$$ , the pair $$(25 124 ...
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1answer
73 views

Can we find such triples systematically , or is it a “lucky strike”?

Here : https://en.wikipedia.org/wiki/Hall%27s_conjecture the so-called Hall-conjecture is formulated and an example is given that would require a very small constant in the Hall-conjecture. It ...
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0answers
128 views

Have the equations $y^2=x^3+n!$ and $y^2=x^3-n!$ infinitely many integral solutions?

The Mordell curve $$y^2=x^3+k$$ is known to have finite many solutions for every integer $k\ne 0$. But is it known whether there are infinite many natural numbers $n$ such that $y^2=x^3-n!$ and/or $y^...
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1answer
88 views

Are there polynomials $f,g$ with $deg(f^2-g^3)=1$?

Define $S$ to be the set of integers $k$, such that the Mordell curve $$y^2=x^3+k$$ has at least one integral point. I would like to get an idea of the structure of $S$. A first step would be to find ...
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1answer
224 views

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 ...
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2answers
138 views

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-...
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0answers
81 views

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 ...
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1answer
89 views

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 ...
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1answer
52 views

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) ...
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1answer
75 views

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 ...
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1answer
41 views

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 ...
1
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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 ...
4
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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 ...
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2answers
203 views

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 ...
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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 ...
5
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1answer
224 views

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 $...
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2answers
264 views

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 $x^3 = 7y^3 + 6 y^2+2 y$?...
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1answer
274 views

Solving the Mordell equation $y^2 = x^3 − 2$; what would be a general strategy?

I am looking at the solution provided in my lecture notes for solving this particular Mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3.$$ In the ...
3
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1answer
66 views

Bounding $x^2+6x$ between consecutive cubes when solving $y^3=x^2+6x$

I am familiar with the method of bounding a polynomial between consecutive squares to prove it is not a square. For example, this method can prove $y^2=x^2+x+1$ has no solutions since $x^2<x^2+x+1&...
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0answers
248 views

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 ...
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0answers
507 views

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 ...
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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)
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1answer
214 views

$x^3-9=y^2$ 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
243 views

The diophantine equation $y^2=x^3+7$ has no solutions

In my lecture notes there is the following example: The diophantine equation $y^2=x^3+7$ has no solutions. Proof: If the equation would have a solution, let $(x_0, y_0)$, $y_0^2=x_0^3+7$, then $...
5
votes
1answer
854 views

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 ...
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1answer
438 views

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 ...
2
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1answer
125 views

Solve $y^2=x^3-4$ in $x,y\in \mathbb{Z}$

I am having trouble solving the diophantine equation given in the title. This is how far I came: We can factor in $\mathbb{Z}[i]$ $y^2+4=x^3\Rightarrow (y+2i)(y-2i)=x^3$. I want to show now that ...
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0answers
436 views

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 $k$, ...
3
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1answer
233 views

How many duplication formulas exist for the Mordell curve family $Y^2-X^3=c$?

For the Mordell equation $$ Y^2-X^3 = c, $$ Bachet gave a famous duplication formula which translates one rational solution $(x_1,y_1)$ into a second rational solution $(x_2,y_2)$. Réalis gave a ...
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0answers
378 views

How to solve $x^2+11=y^3$?

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
4
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2answers
551 views

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 ...
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1answer
194 views

Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$.

$$\Large\textbf{Problem}$$ Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$, the Mordell-Weil group. $$\Large\textbf{Attempts and ...
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7answers
1k views

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 ...
3
votes
1answer
13k views

The Diophantine equation $x^2 + 2 = y^3$

How to solve the Diophantine equation $x^2 + 2 = y^3$ with $x,y>0$ ? ($x,y$ are integers.)
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2answers
213 views

Any software package with Mordell equations implemented?

I admit I have no idea how to tag this post, but I'm looking for a CAS/number theory software package that would implement a decent algorithm for computing the integral solutions to $x^2 = y^3 - k$, ...
5
votes
0answers
129 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in \mathbb{...
5
votes
1answer
2k views

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 ...
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3answers
10k views

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 ...