$x^2 + 3xy + y^2 = n$ Diophantine Equation
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I was wondering if someone could direct me towards information regarding the $x^2 + 3xy +y^2 = n$ diophantine equation. Additionally, is there anything about the general case of these diophantine equations in the form $x^{2k} + 3x^ky^k + y^{2k} = n$?
number-theory diophantine-equations quadratic-forms pell-type-equations
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
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add a comment |
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I was wondering if someone could direct me towards information regarding the $x^2 + 3xy +y^2 = n$ diophantine equation. Additionally, is there anything about the general case of these diophantine equations in the form $x^{2k} + 3x^ky^k + y^{2k} = n$?
number-theory diophantine-equations quadratic-forms pell-type-equations
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
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4
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That's $(2x+3y)^2-5y^2=4n$, so a generalised Pell equation.
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– Lord Shark the Unknown
Dec 24 '18 at 21:08
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While I was looking for solutions to the generalized Pell equation, I encountered solutions to equations such as $x^2 - Dy^2 = pm 4$ if $D equiv 1 pmod{4}$. The solutions to this diophantine equation are given by: $frac{x+ysqrt{5}}{2} = pm left( frac{x_0 + y_0 sqrt{D}}{2} right)^k$. How can this be extrapolated to solve the case where $x^2 - Dy^2 = 4n$?
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– OmicronGamma
Dec 24 '18 at 23:47
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Here's a well known quadratic diophantine equation solver. alpertron.com.ar/QUAD.HTM
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– user25406
Dec 25 '18 at 21:28
add a comment |
$begingroup$
I was wondering if someone could direct me towards information regarding the $x^2 + 3xy +y^2 = n$ diophantine equation. Additionally, is there anything about the general case of these diophantine equations in the form $x^{2k} + 3x^ky^k + y^{2k} = n$?
number-theory diophantine-equations quadratic-forms pell-type-equations
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
$endgroup$
I was wondering if someone could direct me towards information regarding the $x^2 + 3xy +y^2 = n$ diophantine equation. Additionally, is there anything about the general case of these diophantine equations in the form $x^{2k} + 3x^ky^k + y^{2k} = n$?
number-theory diophantine-equations quadratic-forms pell-type-equations
number-theory diophantine-equations quadratic-forms pell-type-equations
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
edited Dec 25 '18 at 0:13
OmicronGamma
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
asked Dec 24 '18 at 21:01
OmicronGammaOmicronGamma
607
607
4
$begingroup$
That's $(2x+3y)^2-5y^2=4n$, so a generalised Pell equation.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:08
$begingroup$
While I was looking for solutions to the generalized Pell equation, I encountered solutions to equations such as $x^2 - Dy^2 = pm 4$ if $D equiv 1 pmod{4}$. The solutions to this diophantine equation are given by: $frac{x+ysqrt{5}}{2} = pm left( frac{x_0 + y_0 sqrt{D}}{2} right)^k$. How can this be extrapolated to solve the case where $x^2 - Dy^2 = 4n$?
$endgroup$
– OmicronGamma
Dec 24 '18 at 23:47
$begingroup$
Here's a well known quadratic diophantine equation solver. alpertron.com.ar/QUAD.HTM
$endgroup$
– user25406
Dec 25 '18 at 21:28
add a comment |
4
$begingroup$
That's $(2x+3y)^2-5y^2=4n$, so a generalised Pell equation.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:08
$begingroup$
While I was looking for solutions to the generalized Pell equation, I encountered solutions to equations such as $x^2 - Dy^2 = pm 4$ if $D equiv 1 pmod{4}$. The solutions to this diophantine equation are given by: $frac{x+ysqrt{5}}{2} = pm left( frac{x_0 + y_0 sqrt{D}}{2} right)^k$. How can this be extrapolated to solve the case where $x^2 - Dy^2 = 4n$?
$endgroup$
– OmicronGamma
Dec 24 '18 at 23:47
$begingroup$
Here's a well known quadratic diophantine equation solver. alpertron.com.ar/QUAD.HTM
$endgroup$
– user25406
Dec 25 '18 at 21:28
4
4
$begingroup$
That's $(2x+3y)^2-5y^2=4n$, so a generalised Pell equation.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:08
$begingroup$
That's $(2x+3y)^2-5y^2=4n$, so a generalised Pell equation.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:08
$begingroup$
While I was looking for solutions to the generalized Pell equation, I encountered solutions to equations such as $x^2 - Dy^2 = pm 4$ if $D equiv 1 pmod{4}$. The solutions to this diophantine equation are given by: $frac{x+ysqrt{5}}{2} = pm left( frac{x_0 + y_0 sqrt{D}}{2} right)^k$. How can this be extrapolated to solve the case where $x^2 - Dy^2 = 4n$?
$endgroup$
– OmicronGamma
Dec 24 '18 at 23:47
$begingroup$
While I was looking for solutions to the generalized Pell equation, I encountered solutions to equations such as $x^2 - Dy^2 = pm 4$ if $D equiv 1 pmod{4}$. The solutions to this diophantine equation are given by: $frac{x+ysqrt{5}}{2} = pm left( frac{x_0 + y_0 sqrt{D}}{2} right)^k$. How can this be extrapolated to solve the case where $x^2 - Dy^2 = 4n$?
$endgroup$
– OmicronGamma
Dec 24 '18 at 23:47
$begingroup$
Here's a well known quadratic diophantine equation solver. alpertron.com.ar/QUAD.HTM
$endgroup$
– user25406
Dec 25 '18 at 21:28
$begingroup$
Here's a well known quadratic diophantine equation solver. alpertron.com.ar/QUAD.HTM
$endgroup$
– user25406
Dec 25 '18 at 21:28
add a comment |
1 Answer
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I recommend Conways's topograph method. Here is the diagram for $u^2 + uv - v^2,$ which is "equivalent" to your form, which is reduced in Zagier's style.
enter image description here
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
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add a comment |
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1 Answer
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$begingroup$
I recommend Conways's topograph method. Here is the diagram for $u^2 + uv - v^2,$ which is "equivalent" to your form, which is reduced in Zagier's style.
enter image description here
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
$endgroup$
add a comment |
$begingroup$
I recommend Conways's topograph method. Here is the diagram for $u^2 + uv - v^2,$ which is "equivalent" to your form, which is reduced in Zagier's style.
enter image description here
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
$endgroup$
add a comment |
$begingroup$
I recommend Conways's topograph method. Here is the diagram for $u^2 + uv - v^2,$ which is "equivalent" to your form, which is reduced in Zagier's style.
enter image description here
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
$endgroup$
I recommend Conways's topograph method. Here is the diagram for $u^2 + uv - v^2,$ which is "equivalent" to your form, which is reduced in Zagier's style.
enter image description here
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
answered Dec 24 '18 at 21:50
Will JagyWill Jagy
105k5103202
105k5103202
add a comment |
add a comment |
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4
$begingroup$
That's $(2x+3y)^2-5y^2=4n$, so a generalised Pell equation.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:08
$begingroup$
While I was looking for solutions to the generalized Pell equation, I encountered solutions to equations such as $x^2 - Dy^2 = pm 4$ if $D equiv 1 pmod{4}$. The solutions to this diophantine equation are given by: $frac{x+ysqrt{5}}{2} = pm left( frac{x_0 + y_0 sqrt{D}}{2} right)^k$. How can this be extrapolated to solve the case where $x^2 - Dy^2 = 4n$?
$endgroup$
– OmicronGamma
Dec 24 '18 at 23:47
$begingroup$
Here's a well known quadratic diophantine equation solver. alpertron.com.ar/QUAD.HTM
$endgroup$
– user25406
Dec 25 '18 at 21:28