Finding integer solution to a quadratic equation in two unknowns [on hold]How do I solve a linear Diophantine equation with three unknowns?Linear Diophantine equation - Find all integer solutionsDiophantine equation has at least $k$ positive integer solutionsIs there any solution to this quadratic Diophantine equation?Finding integer solution of congruence equationIs there any solution to this quadratic Diophantine 3 variables equation?Integer solution to linear equationHelp finding integer solutions of equation.When can we solve a diophantine equation with degree $2$ in $3$ unknowns completely?Solve Quadratic diophantine equation in two unknowns.

I am the light that shines in the dark

How to educate team mate to take screenshots for bugs with out unwanted stuff

Why restrict private health insurance?

Can multiple states demand income tax from an LLC?

How does learning spells work when leveling a multiclass character?

Where is the License file location for Identity Server in Sitecore 9.1?

How can I portion out frozen cookie dough?

Does the US political system, in principle, allow for a no-party system?

What would be the most expensive material to an intergalactic society?

If nine coins are tossed, what is the probability that the number of heads is even?

Should I file my taxes? No income, unemployed, but paid 2k in student loan interest

How can I have x-axis ticks that show ticks scaled in powers of ten?

Are small insurances worth it?

Did Amazon pay $0 in taxes last year?

Will the concrete slab in a partially heated shed conduct a lot of heat to the unconditioned area?

Boss Telling direct supervisor I snitched

Was this cameo in Captain Marvel computer generated?

How would an energy-based "projectile" blow up a spaceship?

What the error in writing this equation by latex?

Book where society has been split into 2 with a wall down the middle where one side embraced high tech whereas other side were totally against tech

Is there a math expression equivalent to the conditional ternary operator?

Why isn't P and P/poly trivially the same?

What is 'Log Memory' in Query Store 2017

Why is there an extra space when I type "ls" on the Desktop?



Finding integer solution to a quadratic equation in two unknowns [on hold]


How do I solve a linear Diophantine equation with three unknowns?Linear Diophantine equation - Find all integer solutionsDiophantine equation has at least $k$ positive integer solutionsIs there any solution to this quadratic Diophantine equation?Finding integer solution of congruence equationIs there any solution to this quadratic Diophantine 3 variables equation?Integer solution to linear equationHelp finding integer solutions of equation.When can we solve a diophantine equation with degree $2$ in $3$ unknowns completely?Solve Quadratic diophantine equation in two unknowns.













1












$begingroup$



We have an equation:
$$m^2 = n^2 + m + n + 2018.$$
Find all integer pairs $(m,n)$ satisfying this equation.











share|cite|improve this question









New contributor




BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$



put on hold as off-topic by Théophile, John Omielan, Eevee Trainer, YiFan, zz20s 2 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Théophile, John Omielan, Eevee Trainer, YiFan, zz20s
If this question can be reworded to fit the rules in the help center, please edit the question.











  • 1




    $begingroup$
    Well, if $(m,n)$ is a solutions, integer or not, what is the formula for $m$ in terms of $n$ (or vice versa)? Now which values can to be integers.
    $endgroup$
    – fleablood
    10 hours ago















1












$begingroup$



We have an equation:
$$m^2 = n^2 + m + n + 2018.$$
Find all integer pairs $(m,n)$ satisfying this equation.











share|cite|improve this question









New contributor




BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$



put on hold as off-topic by Théophile, John Omielan, Eevee Trainer, YiFan, zz20s 2 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Théophile, John Omielan, Eevee Trainer, YiFan, zz20s
If this question can be reworded to fit the rules in the help center, please edit the question.











  • 1




    $begingroup$
    Well, if $(m,n)$ is a solutions, integer or not, what is the formula for $m$ in terms of $n$ (or vice versa)? Now which values can to be integers.
    $endgroup$
    – fleablood
    10 hours ago













1












1








1


1



$begingroup$



We have an equation:
$$m^2 = n^2 + m + n + 2018.$$
Find all integer pairs $(m,n)$ satisfying this equation.











share|cite|improve this question









New contributor




BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$





We have an equation:
$$m^2 = n^2 + m + n + 2018.$$
Find all integer pairs $(m,n)$ satisfying this equation.








elementary-number-theory divisibility diophantine-equations






share|cite|improve this question









New contributor




BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 9 hours ago









greedoid

46.4k1160118




46.4k1160118






New contributor




BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 10 hours ago









BIDS SalvaterraBIDS Salvaterra

121




121




New contributor




BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






BIDS Salvaterra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




put on hold as off-topic by Théophile, John Omielan, Eevee Trainer, YiFan, zz20s 2 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Théophile, John Omielan, Eevee Trainer, YiFan, zz20s
If this question can be reworded to fit the rules in the help center, please edit the question.







put on hold as off-topic by Théophile, John Omielan, Eevee Trainer, YiFan, zz20s 2 hours ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Théophile, John Omielan, Eevee Trainer, YiFan, zz20s
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 1




    $begingroup$
    Well, if $(m,n)$ is a solutions, integer or not, what is the formula for $m$ in terms of $n$ (or vice versa)? Now which values can to be integers.
    $endgroup$
    – fleablood
    10 hours ago












  • 1




    $begingroup$
    Well, if $(m,n)$ is a solutions, integer or not, what is the formula for $m$ in terms of $n$ (or vice versa)? Now which values can to be integers.
    $endgroup$
    – fleablood
    10 hours ago







1




1




$begingroup$
Well, if $(m,n)$ is a solutions, integer or not, what is the formula for $m$ in terms of $n$ (or vice versa)? Now which values can to be integers.
$endgroup$
– fleablood
10 hours ago




$begingroup$
Well, if $(m,n)$ is a solutions, integer or not, what is the formula for $m$ in terms of $n$ (or vice versa)? Now which values can to be integers.
$endgroup$
– fleablood
10 hours ago










3 Answers
3






active

oldest

votes


















12












$begingroup$

Hint $$ (m+n)(m-n)= (m+n)+2018$$



so $$ (m+n)(m-n-1)= 2018$$






share|cite|improve this answer









$endgroup$




















    5












    $begingroup$

    Guide: Write $m=n+k$ for some integer $k$, then $$n^2+2nk+k^2= n^2+2n+k+2018$$



    so $$ n=-k^2+k+2018over 2(k-1)=-kover 2+1009over k-1$$



    If $k$ is odd then there is no solution, so $k= 2s$ so $$2s-1mid 1009$$



    Can you finish?






    share|cite|improve this answer









    $endgroup$




















      1












      $begingroup$

      Simpler start: separating variables to either side gives:
      $$m^2-m=n^2+n+2018$$
      which then factors roughly for the variables as:
      $$m(m-1)=n(n+1)+2018$$



      which since both pairs(m,m-1) and (n,n+1) are consecutive integers, you can divide both sides by two giving:



      $$fracm(m-1)2=fracn(n+1)2+1009$$



      But, $fracy(y+1)2$ is the form of the y-th triangular number, so the solutions are such that 1009 is the difference of two triangular numbers $T_vert m-1 vert$ and $T_vert n vert$ . Solve for n, and m-1 .






      share|cite|improve this answer











      $endgroup$



















        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        12












        $begingroup$

        Hint $$ (m+n)(m-n)= (m+n)+2018$$



        so $$ (m+n)(m-n-1)= 2018$$






        share|cite|improve this answer









        $endgroup$

















          12












          $begingroup$

          Hint $$ (m+n)(m-n)= (m+n)+2018$$



          so $$ (m+n)(m-n-1)= 2018$$






          share|cite|improve this answer









          $endgroup$















            12












            12








            12





            $begingroup$

            Hint $$ (m+n)(m-n)= (m+n)+2018$$



            so $$ (m+n)(m-n-1)= 2018$$






            share|cite|improve this answer









            $endgroup$



            Hint $$ (m+n)(m-n)= (m+n)+2018$$



            so $$ (m+n)(m-n-1)= 2018$$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 10 hours ago









            greedoidgreedoid

            46.4k1160118




            46.4k1160118





















                5












                $begingroup$

                Guide: Write $m=n+k$ for some integer $k$, then $$n^2+2nk+k^2= n^2+2n+k+2018$$



                so $$ n=-k^2+k+2018over 2(k-1)=-kover 2+1009over k-1$$



                If $k$ is odd then there is no solution, so $k= 2s$ so $$2s-1mid 1009$$



                Can you finish?






                share|cite|improve this answer









                $endgroup$

















                  5












                  $begingroup$

                  Guide: Write $m=n+k$ for some integer $k$, then $$n^2+2nk+k^2= n^2+2n+k+2018$$



                  so $$ n=-k^2+k+2018over 2(k-1)=-kover 2+1009over k-1$$



                  If $k$ is odd then there is no solution, so $k= 2s$ so $$2s-1mid 1009$$



                  Can you finish?






                  share|cite|improve this answer









                  $endgroup$















                    5












                    5








                    5





                    $begingroup$

                    Guide: Write $m=n+k$ for some integer $k$, then $$n^2+2nk+k^2= n^2+2n+k+2018$$



                    so $$ n=-k^2+k+2018over 2(k-1)=-kover 2+1009over k-1$$



                    If $k$ is odd then there is no solution, so $k= 2s$ so $$2s-1mid 1009$$



                    Can you finish?






                    share|cite|improve this answer









                    $endgroup$



                    Guide: Write $m=n+k$ for some integer $k$, then $$n^2+2nk+k^2= n^2+2n+k+2018$$



                    so $$ n=-k^2+k+2018over 2(k-1)=-kover 2+1009over k-1$$



                    If $k$ is odd then there is no solution, so $k= 2s$ so $$2s-1mid 1009$$



                    Can you finish?







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 10 hours ago









                    greedoidgreedoid

                    46.4k1160118




                    46.4k1160118





















                        1












                        $begingroup$

                        Simpler start: separating variables to either side gives:
                        $$m^2-m=n^2+n+2018$$
                        which then factors roughly for the variables as:
                        $$m(m-1)=n(n+1)+2018$$



                        which since both pairs(m,m-1) and (n,n+1) are consecutive integers, you can divide both sides by two giving:



                        $$fracm(m-1)2=fracn(n+1)2+1009$$



                        But, $fracy(y+1)2$ is the form of the y-th triangular number, so the solutions are such that 1009 is the difference of two triangular numbers $T_vert m-1 vert$ and $T_vert n vert$ . Solve for n, and m-1 .






                        share|cite|improve this answer











                        $endgroup$

















                          1












                          $begingroup$

                          Simpler start: separating variables to either side gives:
                          $$m^2-m=n^2+n+2018$$
                          which then factors roughly for the variables as:
                          $$m(m-1)=n(n+1)+2018$$



                          which since both pairs(m,m-1) and (n,n+1) are consecutive integers, you can divide both sides by two giving:



                          $$fracm(m-1)2=fracn(n+1)2+1009$$



                          But, $fracy(y+1)2$ is the form of the y-th triangular number, so the solutions are such that 1009 is the difference of two triangular numbers $T_vert m-1 vert$ and $T_vert n vert$ . Solve for n, and m-1 .






                          share|cite|improve this answer











                          $endgroup$















                            1












                            1








                            1





                            $begingroup$

                            Simpler start: separating variables to either side gives:
                            $$m^2-m=n^2+n+2018$$
                            which then factors roughly for the variables as:
                            $$m(m-1)=n(n+1)+2018$$



                            which since both pairs(m,m-1) and (n,n+1) are consecutive integers, you can divide both sides by two giving:



                            $$fracm(m-1)2=fracn(n+1)2+1009$$



                            But, $fracy(y+1)2$ is the form of the y-th triangular number, so the solutions are such that 1009 is the difference of two triangular numbers $T_vert m-1 vert$ and $T_vert n vert$ . Solve for n, and m-1 .






                            share|cite|improve this answer











                            $endgroup$



                            Simpler start: separating variables to either side gives:
                            $$m^2-m=n^2+n+2018$$
                            which then factors roughly for the variables as:
                            $$m(m-1)=n(n+1)+2018$$



                            which since both pairs(m,m-1) and (n,n+1) are consecutive integers, you can divide both sides by two giving:



                            $$fracm(m-1)2=fracn(n+1)2+1009$$



                            But, $fracy(y+1)2$ is the form of the y-th triangular number, so the solutions are such that 1009 is the difference of two triangular numbers $T_vert m-1 vert$ and $T_vert n vert$ . Solve for n, and m-1 .







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited 2 hours ago

























                            answered 8 hours ago









                            Roddy MacPheeRoddy MacPhee

                            24614




                            24614













                                Popular posts from this blog

                                Möglingen Índice Localización Historia Demografía Referencias Enlaces externos Menú de navegación48°53′18″N 9°07′45″E / 48.888333333333, 9.129166666666748°53′18″N 9°07′45″E / 48.888333333333, 9.1291666666667Sitio web oficial Mapa de Möglingen«Gemeinden in Deutschland nach Fläche, Bevölkerung und Postleitzahl am 30.09.2016»Möglingen

                                Virtualbox - Configuration error: Querying “UUID” failed (VERR_CFGM_VALUE_NOT_FOUND)“VERR_SUPLIB_WORLD_WRITABLE” error when trying to installing OS in virtualboxVirtual Box Kernel errorFailed to open a seesion for the virtual machineFailed to open a session for the virtual machineUbuntu 14.04 LTS Virtualbox errorcan't use VM VirtualBoxusing virtualboxI can't run Linux-64 Bit on VirtualBoxUnable to insert the virtual optical disk (VBoxguestaddition) in virtual machine for ubuntu server in win 10VirtuaBox in Ubuntu 18.04 Issues with Win10.ISO Installation

                                Torre de la Isleta Índice Véase también Referencias Bibliografía Enlaces externos Menú de navegación38°25′58″N 0°23′02″O / 38.43277778, -0.3838888938°25′58″N 0°23′02″O / 38.43277778, -0.38388889Torre de la Illeta de l’Horta o Torre Saleta. Base de datos de bienes inmuebles. Patrimonio Cultural. Secretaría de Estado de CulturaFicha BIC Torre de la Illeta de l’Horta. Dirección General de Patrimonio Cultural. Generalitat ValencianaLugares de interés. Ayuntamiento del CampelloTorre de la Isleta en CastillosNet.org