Conditions on those Pell-type equations to admit solutions
I am facing those generalised Pell equations,
$a^2-Db^2=-8$
and
$x^2-Dy^2=8$,
where $D=t^2+8t$ for an odd $t$, $t>2$. In particular, I would like to find the cases where both equations admit (integer) solutions.
I don't know too much about continued fractions and theory of Pell equations, so I tried to work my way isolating $t$. We have (t is positive):
$t=frac{-4y+sqrt{16y^2+x^2-8}}{y^2}$,
and so
$t=-4+frac{sqrt{16y^2+x^2-8}}{y}$.
In the same way we can find
$t=-4+frac{sqrt{16b^2+a^2+8}}{b}$.
But I don't know if this can help me finding a solution. I found some way to compute solutions of a generalised Pell equation given one solution, but I don't think it's useful to me: I don't look for actual solutions for $x$, $y$, $a$, $b$, but for conditions on $t$. I used some online calculator of Pell equations, namely https://www.alpertron.com.ar/QUAD.HTM, to investigate solutions, and I checked that, for odd $tleq 131$, if the second equation admits a solution, the first one doesn't. I tried to study some Pell equation theory, but since my $D$ is a variable I don't know how to compute its continued fraction.
I know the request for $t$ to be odd may seem artificial, but it comes from previous assumptions. I don't know whether we can derive it from the two equations or not. In fact, I'm not very interested in it.
diophantine-equations pell-type-equations
asked Dec 3 '18 at 9:00
Nutella Warrior
63
add a comment |
I am facing those generalised Pell equations,
$a^2-Db^2=-8$
and
$x^2-Dy^2=8$,
where $D=t^2+8t$ for an odd $t$, $t>2$. In particular, I would like to find the cases where both equations admit (integer) solutions.
I don't know too much about continued fractions and theory of Pell equations, so I tried to work my way isolating $t$. We have (t is positive):
$t=frac{-4y+sqrt{16y^2+x^2-8}}{y^2}$,
and so
$t=-4+frac{sqrt{16y^2+x^2-8}}{y}$.
In the same way we can find
$t=-4+frac{sqrt{16b^2+a^2+8}}{b}$.
But I don't know if this can help me finding a solution. I found some way to compute solutions of a generalised Pell equation given one solution, but I don't think it's useful to me: I don't look for actual solutions for $x$, $y$, $a$, $b$, but for conditions on $t$. I used some online calculator of Pell equations, namely https://www.alpertron.com.ar/QUAD.HTM, to investigate solutions, and I checked that, for odd $tleq 131$, if the second equation admits a solution, the first one doesn't. I tried to study some Pell equation theory, but since my $D$ is a variable I don't know how to compute its continued fraction.
I know the request for $t$ to be odd may seem artificial, but it comes from previous assumptions. I don't know whether we can derive it from the two equations or not. In fact, I'm not very interested in it.
diophantine-equations pell-type-equations
asked Dec 3 '18 at 9:00
Nutella Warrior
63
Do you want to find solutions given $t$, or do you want to find all $t$ for which simultaneous solutions exist?
– Servaes
Dec 3 '18 at 11:50
1
I would like to find all $t$ for which simultaneous solutions exist
– Nutella Warrior
Dec 3 '18 at 13:56
add a comment |
I am facing those generalised Pell equations,
$a^2-Db^2=-8$
and
$x^2-Dy^2=8$,
where $D=t^2+8t$ for an odd $t$, $t>2$. In particular, I would like to find the cases where both equations admit (integer) solutions.
I don't know too much about continued fractions and theory of Pell equations, so I tried to work my way isolating $t$. We have (t is positive):
$t=frac{-4y+sqrt{16y^2+x^2-8}}{y^2}$,
and so
$t=-4+frac{sqrt{16y^2+x^2-8}}{y}$.
In the same way we can find
$t=-4+frac{sqrt{16b^2+a^2+8}}{b}$.
But I don't know if this can help me finding a solution. I found some way to compute solutions of a generalised Pell equation given one solution, but I don't think it's useful to me: I don't look for actual solutions for $x$, $y$, $a$, $b$, but for conditions on $t$. I used some online calculator of Pell equations, namely https://www.alpertron.com.ar/QUAD.HTM, to investigate solutions, and I checked that, for odd $tleq 131$, if the second equation admits a solution, the first one doesn't. I tried to study some Pell equation theory, but since my $D$ is a variable I don't know how to compute its continued fraction.
I know the request for $t$ to be odd may seem artificial, but it comes from previous assumptions. I don't know whether we can derive it from the two equations or not. In fact, I'm not very interested in it.
diophantine-equations pell-type-equations
asked Dec 3 '18 at 9:00
Nutella Warrior
63
I am facing those generalised Pell equations,
$a^2-Db^2=-8$
and
$x^2-Dy^2=8$,
where $D=t^2+8t$ for an odd $t$, $t>2$. In particular, I would like to find the cases where both equations admit (integer) solutions.
I don't know too much about continued fractions and theory of Pell equations, so I tried to work my way isolating $t$. We have (t is positive):
$t=frac{-4y+sqrt{16y^2+x^2-8}}{y^2}$,
and so
$t=-4+frac{sqrt{16y^2+x^2-8}}{y}$.
In the same way we can find
$t=-4+frac{sqrt{16b^2+a^2+8}}{b}$.
But I don't know if this can help me finding a solution. I found some way to compute solutions of a generalised Pell equation given one solution, but I don't think it's useful to me: I don't look for actual solutions for $x$, $y$, $a$, $b$, but for conditions on $t$. I used some online calculator of Pell equations, namely https://www.alpertron.com.ar/QUAD.HTM, to investigate solutions, and I checked that, for odd $tleq 131$, if the second equation admits a solution, the first one doesn't. I tried to study some Pell equation theory, but since my $D$ is a variable I don't know how to compute its continued fraction.
I know the request for $t$ to be odd may seem artificial, but it comes from previous assumptions. I don't know whether we can derive it from the two equations or not. In fact, I'm not very interested in it.
diophantine-equations pell-type-equations
diophantine-equations pell-type-equations
asked Dec 3 '18 at 9:00
Nutella Warrior
63
asked Dec 3 '18 at 9:00
Nutella Warrior
63
edited Dec 3 '18 at 10:33
asked Dec 3 '18 at 9:00
Nutella Warrior
63
asked Dec 3 '18 at 9:00
Nutella Warrior
63
asked Dec 3 '18 at 9:00
Nutella Warrior
63
63
Do you want to find solutions given $t$, or do you want to find all $t$ for which simultaneous solutions exist?
– Servaes
Dec 3 '18 at 11:50
1
I would like to find all $t$ for which simultaneous solutions exist
– Nutella Warrior
Dec 3 '18 at 13:56
add a comment |
Do you want to find solutions given $t$, or do you want to find all $t$ for which simultaneous solutions exist?
– Servaes
Dec 3 '18 at 11:50
1
I would like to find all $t$ for which simultaneous solutions exist
– Nutella Warrior
Dec 3 '18 at 13:56
Do you want to find solutions given $t$, or do you want to find all $t$ for which simultaneous solutions exist?
– Servaes
Dec 3 '18 at 11:50
Do you want to find solutions given $t$, or do you want to find all $t$ for which simultaneous solutions exist?
– Servaes
Dec 3 '18 at 11:50
1
1
I would like to find all $t$ for which simultaneous solutions exist
– Nutella Warrior
Dec 3 '18 at 13:56
I would like to find all $t$ for which simultaneous solutions exist
– Nutella Warrior
Dec 3 '18 at 13:56
add a comment |
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Do you want to find solutions given $t$, or do you want to find all $t$ for which simultaneous solutions exist?
– Servaes
Dec 3 '18 at 11:50
1
I would like to find all $t$ for which simultaneous solutions exist
– Nutella Warrior
Dec 3 '18 at 13:56