Mixing
Elliptic Curves (equation)
0
$begingroup$
Is there a specific method to solve equations like $ y^2 = x^3 - n^2x $ ?
I was reading an example about $ y^2 = x^3 - 49x $ and it says that a solution is $ (-frac{63}{16},frac{735}{64}) $ .
How did he find it?
Thanks
In some papers I saw that if I find a solution $ (x_{0},y_{0}) $ and replace $ x $ with $ x-x_{0} $ and $ y $ with $ y=kx+y_{0} $ I will find all rational solutions.
number-theory elliptic-curves
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
$endgroup$
add a comment |
0
$begingroup$
Is there a specific method to solve equations like $ y^2 = x^3 - n^2x $ ?
I was reading an example about $ y^2 = x^3 - 49x $ and it says that a solution is $ (-frac{63}{16},frac{735}{64}) $ .
How did he find it?
Thanks
In some papers I saw that if I find a solution $ (x_{0},y_{0}) $ and replace $ x $ with $ x-x_{0} $ and $ y $ with $ y=kx+y_{0} $ I will find all rational solutions.
number-theory elliptic-curves
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
$endgroup$
$begingroup$
I don't understand what you mean by "solving" such an equation. In what sense does $left(-frac{63}{16},frac{735}{64}right)$ "solve" $y^2= x^3- 49x$?
$endgroup$
– user247327
Jan 27 at 13:33
$begingroup$
I want to know if there is a way to find all rational points $ (x,y) $ giving that $ y^2= x^3 -49x $ or $ y^2= x^3 - n^2x $ in general. This example is from "Arithmetic Progressions of Three squares" by Keith Conrad. The point I wrote verifies the equation
$endgroup$
– Dr.Mathematics
Jan 27 at 13:40
$begingroup$
Have a look at Algorithms for finding rational points on an elliptic curve? from Mathoverflow.
$endgroup$
– kelalaka
Jan 27 at 13:57
$begingroup$
The last paragraph sounds really strange. But, are you familiar with the chord-tangent method of finding more points on an elliptic curve?
$endgroup$
– Jyrki Lahtonen
Jan 27 at 14:38
$begingroup$
@Dr.Mathematics : Do you understand why points of finite order ($E(mathbb{Q})_{tors}$) are very special rational points ? The method to find both are very different. Your previous question is referencing everything you need for $E(mathbb{Q})_{tors}$.
$endgroup$
– reuns
Jan 27 at 17:10
add a comment |
0
0
0
$begingroup$
Is there a specific method to solve equations like $ y^2 = x^3 - n^2x $ ?
I was reading an example about $ y^2 = x^3 - 49x $ and it says that a solution is $ (-frac{63}{16},frac{735}{64}) $ .
How did he find it?
Thanks
In some papers I saw that if I find a solution $ (x_{0},y_{0}) $ and replace $ x $ with $ x-x_{0} $ and $ y $ with $ y=kx+y_{0} $ I will find all rational solutions.
number-theory elliptic-curves
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
$endgroup$
Is there a specific method to solve equations like $ y^2 = x^3 - n^2x $ ?
I was reading an example about $ y^2 = x^3 - 49x $ and it says that a solution is $ (-frac{63}{16},frac{735}{64}) $ .
How did he find it?
Thanks
In some papers I saw that if I find a solution $ (x_{0},y_{0}) $ and replace $ x $ with $ x-x_{0} $ and $ y $ with $ y=kx+y_{0} $ I will find all rational solutions.
number-theory elliptic-curves
number-theory elliptic-curves
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
edited Jan 27 at 13:18
Jyrki Lahtonen
110k13171386
110k13171386
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
asked Jan 27 at 13:15
Dr.MathematicsDr.Mathematics
466
466
$begingroup$
I don't understand what you mean by "solving" such an equation. In what sense does $left(-frac{63}{16},frac{735}{64}right)$ "solve" $y^2= x^3- 49x$?
$endgroup$
– user247327
Jan 27 at 13:33
$begingroup$
I want to know if there is a way to find all rational points $ (x,y) $ giving that $ y^2= x^3 -49x $ or $ y^2= x^3 - n^2x $ in general. This example is from "Arithmetic Progressions of Three squares" by Keith Conrad. The point I wrote verifies the equation
$endgroup$
– Dr.Mathematics
Jan 27 at 13:40
$begingroup$
Have a look at Algorithms for finding rational points on an elliptic curve? from Mathoverflow.
$endgroup$
– kelalaka
Jan 27 at 13:57
$begingroup$
The last paragraph sounds really strange. But, are you familiar with the chord-tangent method of finding more points on an elliptic curve?
$endgroup$
– Jyrki Lahtonen
Jan 27 at 14:38
$begingroup$
@Dr.Mathematics : Do you understand why points of finite order ($E(mathbb{Q})_{tors}$) are very special rational points ? The method to find both are very different. Your previous question is referencing everything you need for $E(mathbb{Q})_{tors}$.
$endgroup$
– reuns
Jan 27 at 17:10
add a comment |
$begingroup$
I don't understand what you mean by "solving" such an equation. In what sense does $left(-frac{63}{16},frac{735}{64}right)$ "solve" $y^2= x^3- 49x$?
$endgroup$
– user247327
Jan 27 at 13:33
$begingroup$
I want to know if there is a way to find all rational points $ (x,y) $ giving that $ y^2= x^3 -49x $ or $ y^2= x^3 - n^2x $ in general. This example is from "Arithmetic Progressions of Three squares" by Keith Conrad. The point I wrote verifies the equation
$endgroup$
– Dr.Mathematics
Jan 27 at 13:40
$begingroup$
Have a look at Algorithms for finding rational points on an elliptic curve? from Mathoverflow.
$endgroup$
– kelalaka
Jan 27 at 13:57
$begingroup$
The last paragraph sounds really strange. But, are you familiar with the chord-tangent method of finding more points on an elliptic curve?
$endgroup$
– Jyrki Lahtonen
Jan 27 at 14:38
$begingroup$
@Dr.Mathematics : Do you understand why points of finite order ($E(mathbb{Q})_{tors}$) are very special rational points ? The method to find both are very different. Your previous question is referencing everything you need for $E(mathbb{Q})_{tors}$.
$endgroup$
– reuns
Jan 27 at 17:10
$begingroup$
I don't understand what you mean by "solving" such an equation. In what sense does $left(-frac{63}{16},frac{735}{64}right)$ "solve" $y^2= x^3- 49x$?
$endgroup$
– user247327
Jan 27 at 13:33
$begingroup$
I don't understand what you mean by "solving" such an equation. In what sense does $left(-frac{63}{16},frac{735}{64}right)$ "solve" $y^2= x^3- 49x$?
$endgroup$
– user247327
Jan 27 at 13:33
$begingroup$
I want to know if there is a way to find all rational points $ (x,y) $ giving that $ y^2= x^3 -49x $ or $ y^2= x^3 - n^2x $ in general. This example is from "Arithmetic Progressions of Three squares" by Keith Conrad. The point I wrote verifies the equation
$endgroup$
– Dr.Mathematics
Jan 27 at 13:40
$begingroup$
I want to know if there is a way to find all rational points $ (x,y) $ giving that $ y^2= x^3 -49x $ or $ y^2= x^3 - n^2x $ in general. This example is from "Arithmetic Progressions of Three squares" by Keith Conrad. The point I wrote verifies the equation
$endgroup$
– Dr.Mathematics
Jan 27 at 13:40
$begingroup$
Have a look at Algorithms for finding rational points on an elliptic curve? from Mathoverflow.
$endgroup$
– kelalaka
Jan 27 at 13:57
$begingroup$
Have a look at Algorithms for finding rational points on an elliptic curve? from Mathoverflow.
$endgroup$
– kelalaka
Jan 27 at 13:57
$begingroup$
The last paragraph sounds really strange. But, are you familiar with the chord-tangent method of finding more points on an elliptic curve?
$endgroup$
– Jyrki Lahtonen
Jan 27 at 14:38
$begingroup$
The last paragraph sounds really strange. But, are you familiar with the chord-tangent method of finding more points on an elliptic curve?
$endgroup$
– Jyrki Lahtonen
Jan 27 at 14:38
$begingroup$
@Dr.Mathematics : Do you understand why points of finite order ($E(mathbb{Q})_{tors}$) are very special rational points ? The method to find both are very different. Your previous question is referencing everything you need for $E(mathbb{Q})_{tors}$.
$endgroup$
– reuns
Jan 27 at 17:10
$begingroup$
@Dr.Mathematics : Do you understand why points of finite order ($E(mathbb{Q})_{tors}$) are very special rational points ? The method to find both are very different. Your previous question is referencing everything you need for $E(mathbb{Q})_{tors}$.
$endgroup$
– reuns
Jan 27 at 17:10
add a comment |
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$begingroup$
I don't understand what you mean by "solving" such an equation. In what sense does $left(-frac{63}{16},frac{735}{64}right)$ "solve" $y^2= x^3- 49x$?
$endgroup$
– user247327
Jan 27 at 13:33
$begingroup$
I want to know if there is a way to find all rational points $ (x,y) $ giving that $ y^2= x^3 -49x $ or $ y^2= x^3 - n^2x $ in general. This example is from "Arithmetic Progressions of Three squares" by Keith Conrad. The point I wrote verifies the equation
$endgroup$
– Dr.Mathematics
Jan 27 at 13:40
$begingroup$
Have a look at Algorithms for finding rational points on an elliptic curve? from Mathoverflow.
$endgroup$
– kelalaka
Jan 27 at 13:57
$begingroup$
The last paragraph sounds really strange. But, are you familiar with the chord-tangent method of finding more points on an elliptic curve?
$endgroup$
– Jyrki Lahtonen
Jan 27 at 14:38
$begingroup$
@Dr.Mathematics : Do you understand why points of finite order ($E(mathbb{Q})_{tors}$) are very special rational points ? The method to find both are very different. Your previous question is referencing everything you need for $E(mathbb{Q})_{tors}$.
$endgroup$
– reuns
Jan 27 at 17:10