Integer equations












2














I have $2$ following problems. Find integer roots of



$$begin{align}
&1)~frac{x+y}{x^2-xy+y^2}=frac3z \
&2)~x^3y^3-4xy^3+y^2+x^2-2y-3=0
end{align}$$



I have no idea to solve them. I try to guess roots of the second, they are $left( -2, 1right), left( 0, -1right), left( 0, 3right), left( 2, 1right) $. Please help me. Thank you very much.






number-theory elementary-number-theory arithmetic diophantine-equations







share|cite|improve this question
























  • These problems seem very interesting. Where did you get them from?
    – Toby Mak
    Dec 8 at 9:39










  • @TobyMak These are my homework.
    – Cglkttca
    Dec 8 at 9:42










  • What class are you taking?
    – Toby Mak
    Dec 8 at 9:42










  • @TobyMak I'm in grade 9.
    – Cglkttca
    Dec 8 at 9:43






  • 1




    What lesson is this that you were given that exercise ?
    – Rebellos
    Dec 8 at 10:03
















2














I have $2$ following problems. Find integer roots of



$$begin{align}
&1)~frac{x+y}{x^2-xy+y^2}=frac3z \
&2)~x^3y^3-4xy^3+y^2+x^2-2y-3=0
end{align}$$



I have no idea to solve them. I try to guess roots of the second, they are $left( -2, 1right), left( 0, -1right), left( 0, 3right), left( 2, 1right) $. Please help me. Thank you very much.






number-theory elementary-number-theory arithmetic diophantine-equations







share|cite|improve this question
























  • These problems seem very interesting. Where did you get them from?
    – Toby Mak
    Dec 8 at 9:39










  • @TobyMak These are my homework.
    – Cglkttca
    Dec 8 at 9:42










  • What class are you taking?
    – Toby Mak
    Dec 8 at 9:42










  • @TobyMak I'm in grade 9.
    – Cglkttca
    Dec 8 at 9:43






  • 1




    What lesson is this that you were given that exercise ?
    – Rebellos
    Dec 8 at 10:03














2












2








2


1





I have $2$ following problems. Find integer roots of



$$begin{align}
&1)~frac{x+y}{x^2-xy+y^2}=frac3z \
&2)~x^3y^3-4xy^3+y^2+x^2-2y-3=0
end{align}$$



I have no idea to solve them. I try to guess roots of the second, they are $left( -2, 1right), left( 0, -1right), left( 0, 3right), left( 2, 1right) $. Please help me. Thank you very much.






number-theory elementary-number-theory arithmetic diophantine-equations







share|cite|improve this question















I have $2$ following problems. Find integer roots of



$$begin{align}
&1)~frac{x+y}{x^2-xy+y^2}=frac3z \
&2)~x^3y^3-4xy^3+y^2+x^2-2y-3=0
end{align}$$



I have no idea to solve them. I try to guess roots of the second, they are $left( -2, 1right), left( 0, -1right), left( 0, 3right), left( 2, 1right) $. Please help me. Thank you very much.






number-theory elementary-number-theory arithmetic diophantine-equations




number-theory elementary-number-theory arithmetic diophantine-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 8 at 15:22









mrtaurho

3,3982932




3,3982932










asked Dec 8 at 9:38









Cglkttca

826




826












  • These problems seem very interesting. Where did you get them from?
    – Toby Mak
    Dec 8 at 9:39










  • @TobyMak These are my homework.
    – Cglkttca
    Dec 8 at 9:42










  • What class are you taking?
    – Toby Mak
    Dec 8 at 9:42










  • @TobyMak I'm in grade 9.
    – Cglkttca
    Dec 8 at 9:43






  • 1




    What lesson is this that you were given that exercise ?
    – Rebellos
    Dec 8 at 10:03


















  • These problems seem very interesting. Where did you get them from?
    – Toby Mak
    Dec 8 at 9:39










  • @TobyMak These are my homework.
    – Cglkttca
    Dec 8 at 9:42










  • What class are you taking?
    – Toby Mak
    Dec 8 at 9:42










  • @TobyMak I'm in grade 9.
    – Cglkttca
    Dec 8 at 9:43






  • 1




    What lesson is this that you were given that exercise ?
    – Rebellos
    Dec 8 at 10:03
















These problems seem very interesting. Where did you get them from?
– Toby Mak
Dec 8 at 9:39




These problems seem very interesting. Where did you get them from?
– Toby Mak
Dec 8 at 9:39












@TobyMak These are my homework.
– Cglkttca
Dec 8 at 9:42




@TobyMak These are my homework.
– Cglkttca
Dec 8 at 9:42












What class are you taking?
– Toby Mak
Dec 8 at 9:42




What class are you taking?
– Toby Mak
Dec 8 at 9:42












@TobyMak I'm in grade 9.
– Cglkttca
Dec 8 at 9:43




@TobyMak I'm in grade 9.
– Cglkttca
Dec 8 at 9:43




1




1




What lesson is this that you were given that exercise ?
– Rebellos
Dec 8 at 10:03




What lesson is this that you were given that exercise ?
– Rebellos
Dec 8 at 10:03










1 Answer
1






active

oldest

votes


















4














Question 1:



begin{align}
& frac{x+y}{x^2-xy+y^2} = frac 3z \
iff & frac{3(x^2-xy+y^2)}{x+y} = z\
iff & 3(x+y) - frac{9xy}{x+y} = z \
iff & frac{9xy}{x+y} = 3x+3y-z \
implies & frac{9}{frac 1x + frac 1y} = 3x+3y-z
end{align}



From the LHS we see that $frac 1x + frac 1y$ must be equal to $pm 1, pm 3, pm 9$ since the RHS is an integer. But of course, since $x,y$ are integers, $frac 1x + frac 1y in [-2,2]$ and it follows that $frac 1x + frac 1y = pm 1$.



Moreover, the only way this can happen is if $x = y = pm 2$.



Hence, the only solutions are $(x,y,z) = (2,2,3)$ and $(x,y,z) = (-2,-2,-3)$.



EDIT:



As pointed out in the comments, I have made the mistake of dividing by $0$, so I have changed the last $iff$ to an $implies$.



If $x=0$, then $z=3y$ so we get the solutions $(x,y,z) = (0,t,3t)$



Similarly when $y=0$ we get the solutions $(x,y,z) = (t,0,3t)$ for any $t in Bbb Z$ with $t neq 0$.






share|cite|improve this answer























  • I got it. Thank you very much.
    – Cglkttca
    Dec 8 at 10:20






  • 1




    It's a nice answer, however, you forgot the cases when x and y are 0.
    – Ankit Kumar
    Dec 8 at 10:29










  • Yes, in this case, we get more solutions
    – Cglkttca
    Dec 8 at 10:34











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030889%2finteger-equations%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









4














Question 1:



begin{align}
& frac{x+y}{x^2-xy+y^2} = frac 3z \
iff & frac{3(x^2-xy+y^2)}{x+y} = z\
iff & 3(x+y) - frac{9xy}{x+y} = z \
iff & frac{9xy}{x+y} = 3x+3y-z \
implies & frac{9}{frac 1x + frac 1y} = 3x+3y-z
end{align}



From the LHS we see that $frac 1x + frac 1y$ must be equal to $pm 1, pm 3, pm 9$ since the RHS is an integer. But of course, since $x,y$ are integers, $frac 1x + frac 1y in [-2,2]$ and it follows that $frac 1x + frac 1y = pm 1$.



Moreover, the only way this can happen is if $x = y = pm 2$.



Hence, the only solutions are $(x,y,z) = (2,2,3)$ and $(x,y,z) = (-2,-2,-3)$.



EDIT:



As pointed out in the comments, I have made the mistake of dividing by $0$, so I have changed the last $iff$ to an $implies$.



If $x=0$, then $z=3y$ so we get the solutions $(x,y,z) = (0,t,3t)$



Similarly when $y=0$ we get the solutions $(x,y,z) = (t,0,3t)$ for any $t in Bbb Z$ with $t neq 0$.






share|cite|improve this answer























  • I got it. Thank you very much.
    – Cglkttca
    Dec 8 at 10:20






  • 1




    It's a nice answer, however, you forgot the cases when x and y are 0.
    – Ankit Kumar
    Dec 8 at 10:29










  • Yes, in this case, we get more solutions
    – Cglkttca
    Dec 8 at 10:34
















4














Question 1:



begin{align}
& frac{x+y}{x^2-xy+y^2} = frac 3z \
iff & frac{3(x^2-xy+y^2)}{x+y} = z\
iff & 3(x+y) - frac{9xy}{x+y} = z \
iff & frac{9xy}{x+y} = 3x+3y-z \
implies & frac{9}{frac 1x + frac 1y} = 3x+3y-z
end{align}



From the LHS we see that $frac 1x + frac 1y$ must be equal to $pm 1, pm 3, pm 9$ since the RHS is an integer. But of course, since $x,y$ are integers, $frac 1x + frac 1y in [-2,2]$ and it follows that $frac 1x + frac 1y = pm 1$.



Moreover, the only way this can happen is if $x = y = pm 2$.



Hence, the only solutions are $(x,y,z) = (2,2,3)$ and $(x,y,z) = (-2,-2,-3)$.



EDIT:



As pointed out in the comments, I have made the mistake of dividing by $0$, so I have changed the last $iff$ to an $implies$.



If $x=0$, then $z=3y$ so we get the solutions $(x,y,z) = (0,t,3t)$



Similarly when $y=0$ we get the solutions $(x,y,z) = (t,0,3t)$ for any $t in Bbb Z$ with $t neq 0$.






share|cite|improve this answer























  • I got it. Thank you very much.
    – Cglkttca
    Dec 8 at 10:20






  • 1




    It's a nice answer, however, you forgot the cases when x and y are 0.
    – Ankit Kumar
    Dec 8 at 10:29










  • Yes, in this case, we get more solutions
    – Cglkttca
    Dec 8 at 10:34














4












4








4






Question 1:



begin{align}
& frac{x+y}{x^2-xy+y^2} = frac 3z \
iff & frac{3(x^2-xy+y^2)}{x+y} = z\
iff & 3(x+y) - frac{9xy}{x+y} = z \
iff & frac{9xy}{x+y} = 3x+3y-z \
implies & frac{9}{frac 1x + frac 1y} = 3x+3y-z
end{align}



From the LHS we see that $frac 1x + frac 1y$ must be equal to $pm 1, pm 3, pm 9$ since the RHS is an integer. But of course, since $x,y$ are integers, $frac 1x + frac 1y in [-2,2]$ and it follows that $frac 1x + frac 1y = pm 1$.



Moreover, the only way this can happen is if $x = y = pm 2$.



Hence, the only solutions are $(x,y,z) = (2,2,3)$ and $(x,y,z) = (-2,-2,-3)$.



EDIT:



As pointed out in the comments, I have made the mistake of dividing by $0$, so I have changed the last $iff$ to an $implies$.



If $x=0$, then $z=3y$ so we get the solutions $(x,y,z) = (0,t,3t)$



Similarly when $y=0$ we get the solutions $(x,y,z) = (t,0,3t)$ for any $t in Bbb Z$ with $t neq 0$.






share|cite|improve this answer














Question 1:



begin{align}
& frac{x+y}{x^2-xy+y^2} = frac 3z \
iff & frac{3(x^2-xy+y^2)}{x+y} = z\
iff & 3(x+y) - frac{9xy}{x+y} = z \
iff & frac{9xy}{x+y} = 3x+3y-z \
implies & frac{9}{frac 1x + frac 1y} = 3x+3y-z
end{align}



From the LHS we see that $frac 1x + frac 1y$ must be equal to $pm 1, pm 3, pm 9$ since the RHS is an integer. But of course, since $x,y$ are integers, $frac 1x + frac 1y in [-2,2]$ and it follows that $frac 1x + frac 1y = pm 1$.



Moreover, the only way this can happen is if $x = y = pm 2$.



Hence, the only solutions are $(x,y,z) = (2,2,3)$ and $(x,y,z) = (-2,-2,-3)$.



EDIT:



As pointed out in the comments, I have made the mistake of dividing by $0$, so I have changed the last $iff$ to an $implies$.



If $x=0$, then $z=3y$ so we get the solutions $(x,y,z) = (0,t,3t)$



Similarly when $y=0$ we get the solutions $(x,y,z) = (t,0,3t)$ for any $t in Bbb Z$ with $t neq 0$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 8 at 10:37

























answered Dec 8 at 10:16









glowstonetrees

2,285317




2,285317












  • I got it. Thank you very much.
    – Cglkttca
    Dec 8 at 10:20






  • 1




    It's a nice answer, however, you forgot the cases when x and y are 0.
    – Ankit Kumar
    Dec 8 at 10:29










  • Yes, in this case, we get more solutions
    – Cglkttca
    Dec 8 at 10:34


















  • I got it. Thank you very much.
    – Cglkttca
    Dec 8 at 10:20






  • 1




    It's a nice answer, however, you forgot the cases when x and y are 0.
    – Ankit Kumar
    Dec 8 at 10:29










  • Yes, in this case, we get more solutions
    – Cglkttca
    Dec 8 at 10:34
















I got it. Thank you very much.
– Cglkttca
Dec 8 at 10:20




I got it. Thank you very much.
– Cglkttca
Dec 8 at 10:20




1




1




It's a nice answer, however, you forgot the cases when x and y are 0.
– Ankit Kumar
Dec 8 at 10:29




It's a nice answer, however, you forgot the cases when x and y are 0.
– Ankit Kumar
Dec 8 at 10:29












Yes, in this case, we get more solutions
– Cglkttca
Dec 8 at 10:34




Yes, in this case, we get more solutions
– Cglkttca
Dec 8 at 10:34


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030889%2finteger-equations%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

Bressuire

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more

Cabo Verde

Image
  Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
Read more

Gyllenstierna

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more