How to find out number of solutions for equations of type $a+b^2+c^3+d^4 le X$? [closed]












0












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Given the value of 'X' how do we calculate the number of solutions of the equation
$$a+b^2+c^3+d^4 le X$$
where $a,b,c,d$ are non negative integer values?
What are these type of equations called and the concept of mathematics that deals with these type of equations?






polynomials systems-of-equations







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closed as off-topic by Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos Jan 26 at 6:54


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." – Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.





















    0












    $begingroup$


    Given the value of 'X' how do we calculate the number of solutions of the equation
    $$a+b^2+c^3+d^4 le X$$
    where $a,b,c,d$ are non negative integer values?
    What are these type of equations called and the concept of mathematics that deals with these type of equations?






    polynomials systems-of-equations







    share|cite|improve this question











    $endgroup$



    closed as off-topic by Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos Jan 26 at 6:54


    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." – Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos

    If this question can be reworded to fit the rules in the help center, please edit the question.



















      0












      0








      0


      1



      $begingroup$


      Given the value of 'X' how do we calculate the number of solutions of the equation
      $$a+b^2+c^3+d^4 le X$$
      where $a,b,c,d$ are non negative integer values?
      What are these type of equations called and the concept of mathematics that deals with these type of equations?






      polynomials systems-of-equations







      share|cite|improve this question











      $endgroup$




      Given the value of 'X' how do we calculate the number of solutions of the equation
      $$a+b^2+c^3+d^4 le X$$
      where $a,b,c,d$ are non negative integer values?
      What are these type of equations called and the concept of mathematics that deals with these type of equations?






      polynomials systems-of-equations




      polynomials systems-of-equations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 25 at 12:16









      Robert Z

      101k1070142




      101k1070142










      asked Jan 25 at 12:11









      NoobScriptNoobScript

      33




      33




      closed as off-topic by Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos Jan 26 at 6:54


      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." – Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos

      If this question can be reworded to fit the rules in the help center, please edit the question.







      closed as off-topic by Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos Jan 26 at 6:54


      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." – Namaste, Adrian Keister, Leucippus, Cesareo, José Carlos Santos

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          1 Answer
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          $begingroup$

          I don't know anything about the background of these equations, but finding general solutions can be done by induction on the degree of the "polynomial".



          First consider $aleq X$, clearly this has $X+1$ solutions.
          Now consider $a+b^{2}leq X$ which is equivalent to $aleq X-b^{2}$. Then $bin{0,...,lfloorsqrt{X}rfloor}$, so there are
          $$sum^{lfloorsqrt{X}rfloor}_{i=0}X-i^{2}+1$$
          solutions.



          Continue this process to find the solution to the "polynomial" of degree 4.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, this really helped a lot. :)
            $endgroup$
            – NoobScript
            Jan 26 at 2:23


















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          I don't know anything about the background of these equations, but finding general solutions can be done by induction on the degree of the "polynomial".



          First consider $aleq X$, clearly this has $X+1$ solutions.
          Now consider $a+b^{2}leq X$ which is equivalent to $aleq X-b^{2}$. Then $bin{0,...,lfloorsqrt{X}rfloor}$, so there are
          $$sum^{lfloorsqrt{X}rfloor}_{i=0}X-i^{2}+1$$
          solutions.



          Continue this process to find the solution to the "polynomial" of degree 4.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, this really helped a lot. :)
            $endgroup$
            – NoobScript
            Jan 26 at 2:23
















          1












          $begingroup$

          I don't know anything about the background of these equations, but finding general solutions can be done by induction on the degree of the "polynomial".



          First consider $aleq X$, clearly this has $X+1$ solutions.
          Now consider $a+b^{2}leq X$ which is equivalent to $aleq X-b^{2}$. Then $bin{0,...,lfloorsqrt{X}rfloor}$, so there are
          $$sum^{lfloorsqrt{X}rfloor}_{i=0}X-i^{2}+1$$
          solutions.



          Continue this process to find the solution to the "polynomial" of degree 4.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you, this really helped a lot. :)
            $endgroup$
            – NoobScript
            Jan 26 at 2:23














          1












          1








          1





          $begingroup$

          I don't know anything about the background of these equations, but finding general solutions can be done by induction on the degree of the "polynomial".



          First consider $aleq X$, clearly this has $X+1$ solutions.
          Now consider $a+b^{2}leq X$ which is equivalent to $aleq X-b^{2}$. Then $bin{0,...,lfloorsqrt{X}rfloor}$, so there are
          $$sum^{lfloorsqrt{X}rfloor}_{i=0}X-i^{2}+1$$
          solutions.



          Continue this process to find the solution to the "polynomial" of degree 4.






          share|cite|improve this answer









          $endgroup$



          I don't know anything about the background of these equations, but finding general solutions can be done by induction on the degree of the "polynomial".



          First consider $aleq X$, clearly this has $X+1$ solutions.
          Now consider $a+b^{2}leq X$ which is equivalent to $aleq X-b^{2}$. Then $bin{0,...,lfloorsqrt{X}rfloor}$, so there are
          $$sum^{lfloorsqrt{X}rfloor}_{i=0}X-i^{2}+1$$
          solutions.



          Continue this process to find the solution to the "polynomial" of degree 4.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 25 at 15:01









          Floris ClaassensFloris Claassens

          1,02716




          1,02716












          • $begingroup$
            Thank you, this really helped a lot. :)
            $endgroup$
            – NoobScript
            Jan 26 at 2:23


















          • $begingroup$
            Thank you, this really helped a lot. :)
            $endgroup$
            – NoobScript
            Jan 26 at 2:23
















          $begingroup$
          Thank you, this really helped a lot. :)
          $endgroup$
          – NoobScript
          Jan 26 at 2:23




          $begingroup$
          Thank you, this really helped a lot. :)
          $endgroup$
          – NoobScript
          Jan 26 at 2:23



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