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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3023809%2fconditions-on-those-pell-type-equations-to-admit-solutions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3023809%2fconditions-on-those-pell-type-equations-to-admit-solutions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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