Diophantine equation 3^a+1=3^b+5^c Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Transforming a Diophantine equation to an elliptic curveNon-negative integer solutions of a single Linear Diophantine EquationDiophantine problemDoes the following Diophantine equation have nontrivial rational solutions?Help with this Diophantine equationHelp with this system of Diophantine equationsThe Theory of Transfinite Diophantine EquationsFind a distinct postive integer solution to this $xyzw=504(x^2+y^2+z^2+w^2)$ diophantine equationExponential diophantine equation systemCombination of $k$-powers and divisibility

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Diophantine equation 3^a+1=3^b+5^c



Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Transforming a Diophantine equation to an elliptic curveNon-negative integer solutions of a single Linear Diophantine EquationDiophantine problemDoes the following Diophantine equation have nontrivial rational solutions?Help with this Diophantine equationHelp with this system of Diophantine equationsThe Theory of Transfinite Diophantine EquationsFind a distinct postive integer solution to this $xyzw=504(x^2+y^2+z^2+w^2)$ diophantine equationExponential diophantine equation systemCombination of $k$-powers and divisibility










1












$begingroup$


This is not a research problem, but challenging enough that I've decided to post it in here:



Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$










share|cite|improve this question









$endgroup$
















    1












    $begingroup$


    This is not a research problem, but challenging enough that I've decided to post it in here:



    Determine all triples $(a,b,c)$ of non-negative integers, satisfying
    $$
    1+3^a = 3^b+5^c.
    $$










    share|cite|improve this question









    $endgroup$














      1












      1








      1





      $begingroup$


      This is not a research problem, but challenging enough that I've decided to post it in here:



      Determine all triples $(a,b,c)$ of non-negative integers, satisfying
      $$
      1+3^a = 3^b+5^c.
      $$










      share|cite|improve this question









      $endgroup$




      This is not a research problem, but challenging enough that I've decided to post it in here:



      Determine all triples $(a,b,c)$ of non-negative integers, satisfying
      $$
      1+3^a = 3^b+5^c.
      $$







      nt.number-theory diophantine-equations elementary-proofs






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 38 mins ago









      kawakawa

      1707




      1707




















          1 Answer
          1






          active

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          2












          $begingroup$

          I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
          $$
          ap^x bq^y = c+ dp^z q^w
          $$

          has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Lucia, many thanks for the paper.
            $endgroup$
            – kawa
            12 mins ago











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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
          $$
          ap^x bq^y = c+ dp^z q^w
          $$

          has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Lucia, many thanks for the paper.
            $endgroup$
            – kawa
            12 mins ago















          2












          $begingroup$

          I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
          $$
          ap^x bq^y = c+ dp^z q^w
          $$

          has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Lucia, many thanks for the paper.
            $endgroup$
            – kawa
            12 mins ago













          2












          2








          2





          $begingroup$

          I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
          $$
          ap^x bq^y = c+ dp^z q^w
          $$

          has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.






          share|cite|improve this answer









          $endgroup$



          I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
          $$
          ap^x bq^y = c+ dp^z q^w
          $$

          has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 16 mins ago









          LuciaLucia

          34.9k5151177




          34.9k5151177











          • $begingroup$
            Lucia, many thanks for the paper.
            $endgroup$
            – kawa
            12 mins ago
















          • $begingroup$
            Lucia, many thanks for the paper.
            $endgroup$
            – kawa
            12 mins ago















          $begingroup$
          Lucia, many thanks for the paper.
          $endgroup$
          – kawa
          12 mins ago




          $begingroup$
          Lucia, many thanks for the paper.
          $endgroup$
          – kawa
          12 mins ago

















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