Given $sqrt {m^2+(n+o)^2}$ is int, is it possible that atleast one of $sqrt {o^2+(n+m)^2}$ or $sqrt...
$begingroup$
Given $m,n,o,sqrt {m^2+(n+o)^2}inmathbb N$ and $ole nle m$, is it a guarantee that both of $sqrt {o^2+(n+m)^2},sqrt {n^2+(o+m)^2}$ are irrational?
What I tried:
Firstly, ${m^2+(n+o)^2}le n^2+(o+m)^2le o^2+(n+m)^2$
So, to show $n^2+(o+m)^2$ is not int, its is enough to show $$2o(m-n)le2sqrt{m^2+(n+o)^2}+1$$
But this is didnt show anything useful. Any hints?
Edit: I realized this is same as asking can there exist right angled triangles with unqual hypotnuese such that sum of other two sides is equal
inequality project-euler
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
$endgroup$
add a comment |
$begingroup$
Given $m,n,o,sqrt {m^2+(n+o)^2}inmathbb N$ and $ole nle m$, is it a guarantee that both of $sqrt {o^2+(n+m)^2},sqrt {n^2+(o+m)^2}$ are irrational?
What I tried:
Firstly, ${m^2+(n+o)^2}le n^2+(o+m)^2le o^2+(n+m)^2$
So, to show $n^2+(o+m)^2$ is not int, its is enough to show $$2o(m-n)le2sqrt{m^2+(n+o)^2}+1$$
But this is didnt show anything useful. Any hints?
Edit: I realized this is same as asking can there exist right angled triangles with unqual hypotnuese such that sum of other two sides is equal
inequality project-euler
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
$endgroup$
add a comment |
$begingroup$
Given $m,n,o,sqrt {m^2+(n+o)^2}inmathbb N$ and $ole nle m$, is it a guarantee that both of $sqrt {o^2+(n+m)^2},sqrt {n^2+(o+m)^2}$ are irrational?
What I tried:
Firstly, ${m^2+(n+o)^2}le n^2+(o+m)^2le o^2+(n+m)^2$
So, to show $n^2+(o+m)^2$ is not int, its is enough to show $$2o(m-n)le2sqrt{m^2+(n+o)^2}+1$$
But this is didnt show anything useful. Any hints?
Edit: I realized this is same as asking can there exist right angled triangles with unqual hypotnuese such that sum of other two sides is equal
inequality project-euler
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
$endgroup$
Given $m,n,o,sqrt {m^2+(n+o)^2}inmathbb N$ and $ole nle m$, is it a guarantee that both of $sqrt {o^2+(n+m)^2},sqrt {n^2+(o+m)^2}$ are irrational?
What I tried:
Firstly, ${m^2+(n+o)^2}le n^2+(o+m)^2le o^2+(n+m)^2$
So, to show $n^2+(o+m)^2$ is not int, its is enough to show $$2o(m-n)le2sqrt{m^2+(n+o)^2}+1$$
But this is didnt show anything useful. Any hints?
Edit: I realized this is same as asking can there exist right angled triangles with unqual hypotnuese such that sum of other two sides is equal
inequality project-euler
inequality project-euler
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
asked Dec 13 '18 at 12:02
AnvitAnvit
1,637419
1,637419
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Why no?
Take $(m,n,o)=(3,3,1).$
We obtain:
$$sqrt{n^2+(o+m)^2}=sqrt{3^2+4^2}=5.$$
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
$endgroup$
$begingroup$
And what if we remove the equality?
$endgroup$
– Anvit
Dec 13 '18 at 12:39
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%2f3037913%2fgiven-sqrt-m2no2-is-int-is-it-possible-that-atleast-one-of-sqrt-o%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
$begingroup$
Why no?
Take $(m,n,o)=(3,3,1).$
We obtain:
$$sqrt{n^2+(o+m)^2}=sqrt{3^2+4^2}=5.$$
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
$endgroup$
$begingroup$
And what if we remove the equality?
$endgroup$
– Anvit
Dec 13 '18 at 12:39
add a comment |
$begingroup$
Why no?
Take $(m,n,o)=(3,3,1).$
We obtain:
$$sqrt{n^2+(o+m)^2}=sqrt{3^2+4^2}=5.$$
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
$endgroup$
$begingroup$
And what if we remove the equality?
$endgroup$
– Anvit
Dec 13 '18 at 12:39
add a comment |
$begingroup$
Why no?
Take $(m,n,o)=(3,3,1).$
We obtain:
$$sqrt{n^2+(o+m)^2}=sqrt{3^2+4^2}=5.$$
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
$endgroup$
Why no?
Take $(m,n,o)=(3,3,1).$
We obtain:
$$sqrt{n^2+(o+m)^2}=sqrt{3^2+4^2}=5.$$
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
answered Dec 13 '18 at 12:32
Michael RozenbergMichael Rozenberg
102k1791195
102k1791195
$begingroup$
And what if we remove the equality?
$endgroup$
– Anvit
Dec 13 '18 at 12:39
add a comment |
$begingroup$
And what if we remove the equality?
$endgroup$
– Anvit
Dec 13 '18 at 12:39
$begingroup$
And what if we remove the equality?
$endgroup$
– Anvit
Dec 13 '18 at 12:39
$begingroup$
And what if we remove the equality?
$endgroup$
– Anvit
Dec 13 '18 at 12:39
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.
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%2f3037913%2fgiven-sqrt-m2no2-is-int-is-it-possible-that-atleast-one-of-sqrt-o%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
