Under what conditions on $a,b,c$ is the sum $a^3 + b^3 + c^3$ strictly greater than $(a+b+c)^2$?
$begingroup$
I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.
elementary-number-theory
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
$endgroup$
add a comment |
$begingroup$
I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.
elementary-number-theory
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
$endgroup$
$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24
$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59
add a comment |
$begingroup$
I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.
elementary-number-theory
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
$endgroup$
I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.
elementary-number-theory
elementary-number-theory
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
asked Nov 28 '18 at 18:19
Matheus AndradeMatheus Andrade
1,168417
1,168417
$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24
$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59
add a comment |
$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24
$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59
$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24
$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24
$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59
$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
$endgroup$
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%2f3017497%2funder-what-conditions-on-a-b-c-is-the-sum-a3-b3-c3-strictly-greater-t%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
$begingroup$
Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
$endgroup$
add a comment |
$begingroup$
Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
$endgroup$
add a comment |
$begingroup$
Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
$endgroup$
Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
answered Nov 28 '18 at 19:12
Robert IsraelRobert Israel
319k23209459
319k23209459
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.
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%2f3017497%2funder-what-conditions-on-a-b-c-is-the-sum-a3-b3-c3-strictly-greater-t%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
$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24
$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59