Prove: if $c^2+8 equiv 0$ mod $p$ then $c^3-7c^2-8c$ is a quadratic residue mod $p$.Show that if $m$ in $mathbbZ$ has greatest common divisor $1$ with $21$, then $m^6-1$ is divisible by $63$.Prove that if $ab equiv 1 pmodp$ and $a$ is quadratic residue mod $p$, then so is $b$Show that 7 is a quadratic residue for any prime p of the form 28k + 1 and 28k + 3.If p $equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.If $anotequiv 0modp$ then there are $p-1$ solutions (ordered pairs) to $x^2-y^2equiv amodp$Quadratic residue $p equiv 3 pmod4 $Least prime quadratic residueProving that if a is a quadratic residue mod p, then -a is also a quadratic residue mod p iff $p equiv 1 pmod 4$Weird quadratic residue question.Question regarding quadratic residueFinding a non-quadratic residue mod $p$

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Prove: if $c^2+8 equiv 0$ mod $p$ then $c^3-7c^2-8c$ is a quadratic residue mod $p$.


Show that if $m$ in $mathbbZ$ has greatest common divisor $1$ with $21$, then $m^6-1$ is divisible by $63$.Prove that if $ab equiv 1 pmodp$ and $a$ is quadratic residue mod $p$, then so is $b$Show that 7 is a quadratic residue for any prime p of the form 28k + 1 and 28k + 3.If p $equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.If $anotequiv 0modp$ then there are $p-1$ solutions (ordered pairs) to $x^2-y^2equiv amodp$Quadratic residue $p equiv 3 pmod4 $Least prime quadratic residueProving that if a is a quadratic residue mod p, then -a is also a quadratic residue mod p iff $p equiv 1 pmod 4$Weird quadratic residue question.Question regarding quadratic residueFinding a non-quadratic residue mod $p$













6












$begingroup$


I want to show:




If $c^2+8 equiv 0$ mod $p$ for prime $p>3$, then $c^3-7c^2-8c$ is a quadratic residue mod $p$.




I have calculated that $c^3-7c^2-8c equiv -7c^2-16c equiv 56- 16c equiv 8(7-2c) equiv c^2 (2c -7)$, so it should be enough to see that $2c -7$ is a quadratic residue. What now?










share|cite|improve this question











$endgroup$
















    6












    $begingroup$


    I want to show:




    If $c^2+8 equiv 0$ mod $p$ for prime $p>3$, then $c^3-7c^2-8c$ is a quadratic residue mod $p$.




    I have calculated that $c^3-7c^2-8c equiv -7c^2-16c equiv 56- 16c equiv 8(7-2c) equiv c^2 (2c -7)$, so it should be enough to see that $2c -7$ is a quadratic residue. What now?










    share|cite|improve this question











    $endgroup$














      6












      6








      6





      $begingroup$


      I want to show:




      If $c^2+8 equiv 0$ mod $p$ for prime $p>3$, then $c^3-7c^2-8c$ is a quadratic residue mod $p$.




      I have calculated that $c^3-7c^2-8c equiv -7c^2-16c equiv 56- 16c equiv 8(7-2c) equiv c^2 (2c -7)$, so it should be enough to see that $2c -7$ is a quadratic residue. What now?










      share|cite|improve this question











      $endgroup$




      I want to show:




      If $c^2+8 equiv 0$ mod $p$ for prime $p>3$, then $c^3-7c^2-8c$ is a quadratic residue mod $p$.




      I have calculated that $c^3-7c^2-8c equiv -7c^2-16c equiv 56- 16c equiv 8(7-2c) equiv c^2 (2c -7)$, so it should be enough to see that $2c -7$ is a quadratic residue. What now?







      number-theory elementary-number-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 22 at 9:02







      AJ.

















      asked Mar 22 at 8:46









      AJ.AJ.

      333




      333




















          3 Answers
          3






          active

          oldest

          votes


















          4












          $begingroup$

          $$2c-7equiv2c-7+c^2+8pmod pequiv(c+1)^2$$






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            @AJ. Have you noticed my other answer. That is much more natural derivation
            $endgroup$
            – lab bhattacharjee
            Mar 22 at 9:10


















          3












          $begingroup$

          We have that
          $$(c +1)^2 = c^2 + 2c + 1 = c^2 + 2c + 8 -7 = (c^2 + 8) + (2c - 7) equiv 2c - 7 mod p.$$
          This proves the claim.






          share|cite|improve this answer









          $endgroup$




















            2












            $begingroup$

            Hint:



            As $c^2equiv-8pmod p,$



            $$c(c+1)=c^2+cequiv c-8pmod p$$



            $c(c+1)(c-8)equiv?$



            I believe this is how the problem naturally came into being .






            share|cite|improve this answer









            $endgroup$








            • 1




              $begingroup$
              Not sure about any mistake which has caused the down-vote ! This has been put in a separate post as the approach in the other markedly different.
              $endgroup$
              – lab bhattacharjee
              Mar 22 at 9:21










            • $begingroup$
              Me neither... $(+1)$
              $endgroup$
              – user477343
              Mar 24 at 6:12











            Your Answer





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            3 Answers
            3






            active

            oldest

            votes








            3 Answers
            3






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            4












            $begingroup$

            $$2c-7equiv2c-7+c^2+8pmod pequiv(c+1)^2$$






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              @AJ. Have you noticed my other answer. That is much more natural derivation
              $endgroup$
              – lab bhattacharjee
              Mar 22 at 9:10















            4












            $begingroup$

            $$2c-7equiv2c-7+c^2+8pmod pequiv(c+1)^2$$






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              @AJ. Have you noticed my other answer. That is much more natural derivation
              $endgroup$
              – lab bhattacharjee
              Mar 22 at 9:10













            4












            4








            4





            $begingroup$

            $$2c-7equiv2c-7+c^2+8pmod pequiv(c+1)^2$$






            share|cite|improve this answer









            $endgroup$



            $$2c-7equiv2c-7+c^2+8pmod pequiv(c+1)^2$$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Mar 22 at 8:53









            lab bhattacharjeelab bhattacharjee

            228k15158279




            228k15158279











            • $begingroup$
              @AJ. Have you noticed my other answer. That is much more natural derivation
              $endgroup$
              – lab bhattacharjee
              Mar 22 at 9:10
















            • $begingroup$
              @AJ. Have you noticed my other answer. That is much more natural derivation
              $endgroup$
              – lab bhattacharjee
              Mar 22 at 9:10















            $begingroup$
            @AJ. Have you noticed my other answer. That is much more natural derivation
            $endgroup$
            – lab bhattacharjee
            Mar 22 at 9:10




            $begingroup$
            @AJ. Have you noticed my other answer. That is much more natural derivation
            $endgroup$
            – lab bhattacharjee
            Mar 22 at 9:10











            3












            $begingroup$

            We have that
            $$(c +1)^2 = c^2 + 2c + 1 = c^2 + 2c + 8 -7 = (c^2 + 8) + (2c - 7) equiv 2c - 7 mod p.$$
            This proves the claim.






            share|cite|improve this answer









            $endgroup$

















              3












              $begingroup$

              We have that
              $$(c +1)^2 = c^2 + 2c + 1 = c^2 + 2c + 8 -7 = (c^2 + 8) + (2c - 7) equiv 2c - 7 mod p.$$
              This proves the claim.






              share|cite|improve this answer









              $endgroup$















                3












                3








                3





                $begingroup$

                We have that
                $$(c +1)^2 = c^2 + 2c + 1 = c^2 + 2c + 8 -7 = (c^2 + 8) + (2c - 7) equiv 2c - 7 mod p.$$
                This proves the claim.






                share|cite|improve this answer









                $endgroup$



                We have that
                $$(c +1)^2 = c^2 + 2c + 1 = c^2 + 2c + 8 -7 = (c^2 + 8) + (2c - 7) equiv 2c - 7 mod p.$$
                This proves the claim.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Mar 22 at 8:54









                StudentStudent

                2,1881727




                2,1881727





















                    2












                    $begingroup$

                    Hint:



                    As $c^2equiv-8pmod p,$



                    $$c(c+1)=c^2+cequiv c-8pmod p$$



                    $c(c+1)(c-8)equiv?$



                    I believe this is how the problem naturally came into being .






                    share|cite|improve this answer









                    $endgroup$








                    • 1




                      $begingroup$
                      Not sure about any mistake which has caused the down-vote ! This has been put in a separate post as the approach in the other markedly different.
                      $endgroup$
                      – lab bhattacharjee
                      Mar 22 at 9:21










                    • $begingroup$
                      Me neither... $(+1)$
                      $endgroup$
                      – user477343
                      Mar 24 at 6:12















                    2












                    $begingroup$

                    Hint:



                    As $c^2equiv-8pmod p,$



                    $$c(c+1)=c^2+cequiv c-8pmod p$$



                    $c(c+1)(c-8)equiv?$



                    I believe this is how the problem naturally came into being .






                    share|cite|improve this answer









                    $endgroup$








                    • 1




                      $begingroup$
                      Not sure about any mistake which has caused the down-vote ! This has been put in a separate post as the approach in the other markedly different.
                      $endgroup$
                      – lab bhattacharjee
                      Mar 22 at 9:21










                    • $begingroup$
                      Me neither... $(+1)$
                      $endgroup$
                      – user477343
                      Mar 24 at 6:12













                    2












                    2








                    2





                    $begingroup$

                    Hint:



                    As $c^2equiv-8pmod p,$



                    $$c(c+1)=c^2+cequiv c-8pmod p$$



                    $c(c+1)(c-8)equiv?$



                    I believe this is how the problem naturally came into being .






                    share|cite|improve this answer









                    $endgroup$



                    Hint:



                    As $c^2equiv-8pmod p,$



                    $$c(c+1)=c^2+cequiv c-8pmod p$$



                    $c(c+1)(c-8)equiv?$



                    I believe this is how the problem naturally came into being .







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Mar 22 at 8:57









                    lab bhattacharjeelab bhattacharjee

                    228k15158279




                    228k15158279







                    • 1




                      $begingroup$
                      Not sure about any mistake which has caused the down-vote ! This has been put in a separate post as the approach in the other markedly different.
                      $endgroup$
                      – lab bhattacharjee
                      Mar 22 at 9:21










                    • $begingroup$
                      Me neither... $(+1)$
                      $endgroup$
                      – user477343
                      Mar 24 at 6:12












                    • 1




                      $begingroup$
                      Not sure about any mistake which has caused the down-vote ! This has been put in a separate post as the approach in the other markedly different.
                      $endgroup$
                      – lab bhattacharjee
                      Mar 22 at 9:21










                    • $begingroup$
                      Me neither... $(+1)$
                      $endgroup$
                      – user477343
                      Mar 24 at 6:12







                    1




                    1




                    $begingroup$
                    Not sure about any mistake which has caused the down-vote ! This has been put in a separate post as the approach in the other markedly different.
                    $endgroup$
                    – lab bhattacharjee
                    Mar 22 at 9:21




                    $begingroup$
                    Not sure about any mistake which has caused the down-vote ! This has been put in a separate post as the approach in the other markedly different.
                    $endgroup$
                    – lab bhattacharjee
                    Mar 22 at 9:21












                    $begingroup$
                    Me neither... $(+1)$
                    $endgroup$
                    – user477343
                    Mar 24 at 6:12




                    $begingroup$
                    Me neither... $(+1)$
                    $endgroup$
                    – user477343
                    Mar 24 at 6:12

















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