pell's equation converge
Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase
For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.
algebraic-topology pell-type-equations
asked Dec 9 at 1:54
Wendy
1
add a comment |
Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase
For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.
algebraic-topology pell-type-equations
asked Dec 9 at 1:54
Wendy
1
take a look at en.wikipedia.org/wiki/Pell%27s_equation
– reuns
Dec 9 at 2:03
if you are going to call the fixed coefficient $n,$ you should change the index. Other errors as well. Correct: $$ x_j^2 - n y_j^2 = 1 $$
– Will Jagy
Dec 9 at 2:17
add a comment |
Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase
For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.
algebraic-topology pell-type-equations
asked Dec 9 at 1:54
Wendy
1
Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase
For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.
algebraic-topology pell-type-equations
algebraic-topology pell-type-equations
asked Dec 9 at 1:54
Wendy
1
asked Dec 9 at 1:54
Wendy
1
asked Dec 9 at 1:54
Wendy
1
asked Dec 9 at 1:54
Wendy
1
asked Dec 9 at 1:54
Wendy
1
1
take a look at en.wikipedia.org/wiki/Pell%27s_equation
– reuns
Dec 9 at 2:03
if you are going to call the fixed coefficient $n,$ you should change the index. Other errors as well. Correct: $$ x_j^2 - n y_j^2 = 1 $$
– Will Jagy
Dec 9 at 2:17
add a comment |
take a look at en.wikipedia.org/wiki/Pell%27s_equation
– reuns
Dec 9 at 2:03
if you are going to call the fixed coefficient $n,$ you should change the index. Other errors as well. Correct: $$ x_j^2 - n y_j^2 = 1 $$
– Will Jagy
Dec 9 at 2:17
take a look at en.wikipedia.org/wiki/Pell%27s_equation
– reuns
Dec 9 at 2:03
take a look at en.wikipedia.org/wiki/Pell%27s_equation
– reuns
Dec 9 at 2:03
if you are going to call the fixed coefficient $n,$ you should change the index. Other errors as well. Correct: $$ x_j^2 - n y_j^2 = 1 $$
– Will Jagy
Dec 9 at 2:17
if you are going to call the fixed coefficient $n,$ you should change the index. Other errors as well. Correct: $$ x_j^2 - n y_j^2 = 1 $$
– Will Jagy
Dec 9 at 2:17
add a comment |
1 Answer
1
active
oldest
votes
Let $mathcal{P}_d(mathbb{Z})^+$ be the set of the positive integer solutions to the Pell equation $x^2-dy^2=1$, we have that, if $(a,b)inmathcal{P}_d(mathbb{Z})^+$, then, $a/b$ is a convergent of $sqrt d$, when you square $a/b$ you are getting ridiculously good rational approximations of $d$, remember convergents give you the "best" rational approximations
answered Dec 17 at 2:24
Bruno Andrades
1687
add a comment |
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%2f3031911%2fpells-equation-converge%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Let $mathcal{P}_d(mathbb{Z})^+$ be the set of the positive integer solutions to the Pell equation $x^2-dy^2=1$, we have that, if $(a,b)inmathcal{P}_d(mathbb{Z})^+$, then, $a/b$ is a convergent of $sqrt d$, when you square $a/b$ you are getting ridiculously good rational approximations of $d$, remember convergents give you the "best" rational approximations
answered Dec 17 at 2:24
Bruno Andrades
1687
add a comment |
Let $mathcal{P}_d(mathbb{Z})^+$ be the set of the positive integer solutions to the Pell equation $x^2-dy^2=1$, we have that, if $(a,b)inmathcal{P}_d(mathbb{Z})^+$, then, $a/b$ is a convergent of $sqrt d$, when you square $a/b$ you are getting ridiculously good rational approximations of $d$, remember convergents give you the "best" rational approximations
answered Dec 17 at 2:24
Bruno Andrades
1687
add a comment |
Let $mathcal{P}_d(mathbb{Z})^+$ be the set of the positive integer solutions to the Pell equation $x^2-dy^2=1$, we have that, if $(a,b)inmathcal{P}_d(mathbb{Z})^+$, then, $a/b$ is a convergent of $sqrt d$, when you square $a/b$ you are getting ridiculously good rational approximations of $d$, remember convergents give you the "best" rational approximations
answered Dec 17 at 2:24
Bruno Andrades
1687
Let $mathcal{P}_d(mathbb{Z})^+$ be the set of the positive integer solutions to the Pell equation $x^2-dy^2=1$, we have that, if $(a,b)inmathcal{P}_d(mathbb{Z})^+$, then, $a/b$ is a convergent of $sqrt d$, when you square $a/b$ you are getting ridiculously good rational approximations of $d$, remember convergents give you the "best" rational approximations
answered Dec 17 at 2:24
Bruno Andrades
1687
answered Dec 17 at 2:24
Bruno Andrades
1687
answered Dec 17 at 2:24
Bruno Andrades
1687
answered Dec 17 at 2:24
Bruno Andrades
1687
1687
add a comment |
add a comment |
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%2f3031911%2fpells-equation-converge%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
take a look at en.wikipedia.org/wiki/Pell%27s_equation
– reuns
Dec 9 at 2:03
if you are going to call the fixed coefficient $n,$ you should change the index. Other errors as well. Correct: $$ x_j^2 - n y_j^2 = 1 $$
– Will Jagy
Dec 9 at 2:17