Diophantine equation 3^a+1=3^b+5^c Planned maintenance scheduled April 23, 2019 at 23:30 UTC...



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
$$
1+3^a = 3^b+5^c.
$$






nt.number-theory diophantine-equations elementary-proofs







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
    $$
    1+3^a = 3^b+5^c.
    $$






    nt.number-theory diophantine-equations elementary-proofs







    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
      $$
      1+3^a = 3^b+5^c.
      $$






      nt.number-theory diophantine-equations elementary-proofs







      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




      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 42 mins ago









      kawakawa

      1707




      1707






















          1 Answer
          1






          active

          oldest

          votes


















          4












          $begingroup$

          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
            17 mins ago












          Your Answer








          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "504"
          };
          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%2fmathoverflow.net%2fquestions%2f328619%2fdiophantine-equation-3a1-3b5c%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












          $begingroup$

          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
            17 mins ago
















          4












          $begingroup$

          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
            17 mins ago














          4












          4








          4





          $begingroup$

          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 21 mins ago









          LuciaLucia

          34.9k5151177




          34.9k5151177












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


















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
















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




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


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to MathOverflow!


          • 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.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f328619%2fdiophantine-equation-3a1-3b5c%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

          What is the “three and three hundred thousand syndrome”?Who wrote the book Arena?What five creatures were...

          Image
          What happens when the centripetal force is equal and opposite to the centrifugal force? Why is "la Gestapo" feminine? Nested Dynamic SOQL Query How are passwords stolen from companies if they only store hashes? Exposing a company lying about themselves in a tightly knit industry: Is my career at risk on the long run? Can "few" be used as a subject? If so, what is the rule? Why are there no stars visible in cislunar space? categorizing a variable turns it from insignificant to significant Do I need an EFI partition for each 18.04 ubuntu I have on my HD? What are the consequences of changing the number of hours in ...
          Read more

          Gersau Kjelder | Navigasjonsmeny46°59′0″N 8°31′0″E46°59′0″N...

          Image
          Tidlegare republikkarSkipingar i 1433Kommunar i SchwyzDistrikt i SchwyzNedleggingar i 1798Stader ved Vierwaldstättersee kommunekantonenSchwyzSveitskm²VierwaldstätterseeLauerz Gersau Kommunevåpenet til Gersau Plassering Styresmakter Land   Sveits Kanton kantonen Schwyz Distrikt Gersau Språk tysk Geografi Flatevidd  - kommune 23,70 km² Innbyggjarar  - kommune (desember 2010)    - folketettleik 2 094   88,4 /km² Koordinatar 46°59′0″N 8°31′0″E  Koordinatar: 46°59′0″N 8°31′0″E  Høgd over havet 435 moh Diverse annan informasjon Postnummer 6442 Nettstad: www.gersau.ch Freistaat und Republik Gersau Fristaten og republikken Gersau By-republikk og alliert protektorat i Det gamle sveitsiske eidsforundet ←   ← 1433–1798 1814–1818 →   → Hovudstad Gersau Styreform Republikk Historisk periode Tidleg moderne tid  - Kjøpte fridom frå ...
          Read more

          Hestehale Innhaldsliste Hestehale på kvinner | Hestehale på menn | Galleri | Sjå òg |...

          Image
          Frisyrar frisyrehårhårstrikkhårklemmerøyrethesthovudverkeuropeisk motejobbtreningfletterparykkarungdomsopprørmotkulturmusikararmotorsykkelentusiastarpolitietpolitiskforretningverksemdfiskeflette Hestehale sett bakfrå. Hestehale på ein mannleg indianarleiar. Hestehale er ein frisyre der ein samlar det meste eller alt av langt hår [1] i bakhovudet med ein hårstrikk eller liknande. Avhengig av mote kan det òg vere vanleg å sette opp hestehalen med hårklemmer eller å ha hestehalen hengande på den eine sida av hovudet ved å samle håret ved øyret. Hestehale er ein vanleg frisyre for jenter og for gutar med langt hår. Han har fått namnet sitt etter likskapen med halen på ein hest. For stramme hestehaler kan føre til hårtap, [2] i nokre høve også hovudverk. [3] Innhaldsliste 1 Hestehale på kvinner 2 Hestehale på menn 3 Galleri 4 Sjå òg 5 Bakgrunnsstoff 6 Kjelder Hestehale på kvinner | Ungjente med hestehale som held håret unna ansiktet...
          Read more