3
votes
1answer
54 views

Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
3
votes
1answer
64 views

rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$ y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)? $$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
4
votes
2answers
101 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
3
votes
1answer
79 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
0
votes
0answers
46 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
10
votes
0answers
209 views

Help with $x^4+y^4+z^4 = 1$?

There are exactly 20 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{10}$. Noam Elkies (who kindly answered Question 1 below) showed that the form, $$(x+y)^4+(x-y)^4+z^4 = ...
1
vote
1answer
104 views

Does this equation have a rational point? (Elliptic curve?)

Can someone check pls if, $$852 + 3017 x - 1104 x^2 + 2009 x^3 - 3362 x^4=y^2$$ has a rational point? (This arose in an equal sums of like powers problem.) P.S. I've checked $x=p/q$ for ...
5
votes
1answer
197 views

On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication ...
5
votes
2answers
188 views

Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with Sage

I'm trying to use Sage on-line,but I meet some trouble with the code of it. I want to find the rational points on an ellipse curve,such as $$1 + 18 x + 81 x^2 + 44 x^3 = y^2,\tag1$$ I know that ...
21
votes
3answers
603 views

Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt's inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies ...
3
votes
1answer
269 views

Find all integer solutions to $x^2+4=y^3$. [duplicate]

Find all integer solutions to $x^2+4=y^3$. Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
1
vote
0answers
88 views

On the elliptic curve $x^4+y^4 =193z^2$

Given the simultaneous Diophantine equations, $$u^2+v^2=w^2\tag{1}$$ $$x^4+y^4 = (u^6+v^6)t^2\tag{2}$$ the only solutions seem to be for the first Pythagorean triple $u,v,w = 3,4,5$ which yield the ...
10
votes
2answers
117 views

Solve : $ab(a+b)(a-b)=c^2-1$

As we know that $ab(a+b)(a-b)=c^2$ has no integer solution in $Z^+$.However, it seems that $$ab(a+b)(a-b)=c^2-1$$ has infinite positive integer solutions,could you prove it? Here are some of them: ...
6
votes
3answers
273 views

Diophantine equation $x^2 + 32x = y^3$

I am trying to find all solutions to the Diophantine equation $x^2 + 32x = y^3$. I think that the first step is to factorise: $x(x+32)=y^3$. If $x$ is odd, then $x+32$ is also odd. The common ...
4
votes
3answers
176 views

The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
12
votes
2answers
201 views

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube. I found $x=0$, any others ?
1
vote
1answer
91 views

Like Diophantine equation

The equation $x^n - ny^x-nxy$ = $0$ has solution set $(n, x, y) = (1, 1, \frac12), (2, 1, \frac14), (3, 1, \frac16), \ldots$ I would like to know/learn the following (Kindly discuss) 1) If we ...
1
vote
1answer
134 views

Prove that there are no natural numbers, $i, j$ such that $ 3i^2+3i+7=j^3$

I'm not sure if this is true but, I've tried with many different values of $i, j$ and didn't get any contradictions. The question again, here Prove that there are no natural numbers, $i, j$ such ...
0
votes
1answer
106 views

Solutions of $a^2 = b^d -3^c$

The solutions of $a^2 = b^d -3^c$ are in the form $(a, b, c, d) = ((46)27^t, (13)9^t, 6t+4, 3)$. This is done by using calculator. As per my calculator, I have checked some terms, which are satisfied ...
0
votes
0answers
94 views

Pell type equation cum elliptic curve equation

I have seen this equation $y^3 - 3x^2 = p^m$ to determine the solutions. I know this is elliptic curve. I had some knowledge of elliptic curve. But, I was totally upset to determine the solutions of ...
22
votes
1answer
1k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
3
votes
3answers
220 views

Show that the curve $y^2 = x^3 + 2x^2$ has a double point, and find all rational points

Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve. By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$. By implicit ...
2
votes
1answer
228 views

Form of minimal integral Weierstrass equation for elliptic curve over $Q$ with good reduction at $2$ and $3$.

If $E$ is an elliptic curve over $\mathbb{Q}$ which has good reduction at $2$ and $3$, is it always possible to find a minimal integral Weierstrass equation for $E$ of the form $y^2 = x^3 + Ax + B$ ...