Prove that $1^2 cdot 3^2 cdot 5^2cdots (p-2)^2equiv (-1)^{frac{p+1}{2}} pmod p$












0














$1^2 cdot 3^2 cdot 5^2cdots (p-2)^2equiv (-1)^{frac{p+1}{2}}pmod p$



I saw that I can use Wilson's theorem that $(p-1)!equiv -1 pmod p$ and that they change something, they said put on even number $frac{p+1}{2}$ so now you prove it but I do not understand can you help me?






discrete-mathematics







share|cite|improve this question





























    0














    $1^2 cdot 3^2 cdot 5^2cdots (p-2)^2equiv (-1)^{frac{p+1}{2}}pmod p$



    I saw that I can use Wilson's theorem that $(p-1)!equiv -1 pmod p$ and that they change something, they said put on even number $frac{p+1}{2}$ so now you prove it but I do not understand can you help me?






    discrete-mathematics







    share|cite|improve this question



























      0












      0








      0







      $1^2 cdot 3^2 cdot 5^2cdots (p-2)^2equiv (-1)^{frac{p+1}{2}}pmod p$



      I saw that I can use Wilson's theorem that $(p-1)!equiv -1 pmod p$ and that they change something, they said put on even number $frac{p+1}{2}$ so now you prove it but I do not understand can you help me?






      discrete-mathematics







      share|cite|improve this question















      $1^2 cdot 3^2 cdot 5^2cdots (p-2)^2equiv (-1)^{frac{p+1}{2}}pmod p$



      I saw that I can use Wilson's theorem that $(p-1)!equiv -1 pmod p$ and that they change something, they said put on even number $frac{p+1}{2}$ so now you prove it but I do not understand can you help me?






      discrete-mathematics




      discrete-mathematics






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 25 at 20:47









      Jean-Claude Arbaut

      14.7k63464




      14.7k63464










      asked Nov 25 at 20:31









      Marko Škorić

      70310




      70310






















          1 Answer
          1






          active

          oldest

          votes


















          3














          We work in the field $Bbb F_p$ with $p$ elements. (Same as $Bbb Z/p$, for the given prime $p$, i suppose it is a prime.)



          Then
          $$
          begin{aligned}
          -1 &= 1cdot 2cdot 3cdot 4cdot 5cdot 6cdot dots cdot (p-2)cdot(p-1)
          \
          &=1color{red}{cdot (-2)}cdot 3color{red}{cdot (-4)}cdot 5color{red}{cdot (-6)}cdot dots cdot (p-2)color{red}{cdot(-(p-1))}
          color{blue}{cdot(-1)^{(p-1)/2}}
          \
          &=1cdot 3cdot 5cdot dots cdot (p-2)
          color{red}{cdot (-2)cdot (-4)cdot (-6)cdot dots cdot(-(p-1))}
          color{blue}{cdot(-1)^{(p-1)/2}}
          \
          &=1cdot 3cdot 5cdot dots cdot (p-2)
          color{red}{cdot (p-2)cdot (p-4)cdot (p-6)cdot dots cdot(p-(p-1))}
          color{blue}{cdot(-1)^{(p-1)/2}}
          \
          &=(1cdot 3cdot 5cdot dots cdot (p-2))^2
          color{blue}{cdot(-1)^{(p-1)/2}}
          .
          end{aligned}
          $$
          This leads to the answer...






          share|cite|improve this answer





















            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            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%2fmath.stackexchange.com%2fquestions%2f3013352%2fprove-that-12-cdot-32-cdot-52-cdots-p-22-equiv-1-fracp12-pm%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









            3














            We work in the field $Bbb F_p$ with $p$ elements. (Same as $Bbb Z/p$, for the given prime $p$, i suppose it is a prime.)



            Then
            $$
            begin{aligned}
            -1 &= 1cdot 2cdot 3cdot 4cdot 5cdot 6cdot dots cdot (p-2)cdot(p-1)
            \
            &=1color{red}{cdot (-2)}cdot 3color{red}{cdot (-4)}cdot 5color{red}{cdot (-6)}cdot dots cdot (p-2)color{red}{cdot(-(p-1))}
            color{blue}{cdot(-1)^{(p-1)/2}}
            \
            &=1cdot 3cdot 5cdot dots cdot (p-2)
            color{red}{cdot (-2)cdot (-4)cdot (-6)cdot dots cdot(-(p-1))}
            color{blue}{cdot(-1)^{(p-1)/2}}
            \
            &=1cdot 3cdot 5cdot dots cdot (p-2)
            color{red}{cdot (p-2)cdot (p-4)cdot (p-6)cdot dots cdot(p-(p-1))}
            color{blue}{cdot(-1)^{(p-1)/2}}
            \
            &=(1cdot 3cdot 5cdot dots cdot (p-2))^2
            color{blue}{cdot(-1)^{(p-1)/2}}
            .
            end{aligned}
            $$
            This leads to the answer...






            share|cite|improve this answer


























              3














              We work in the field $Bbb F_p$ with $p$ elements. (Same as $Bbb Z/p$, for the given prime $p$, i suppose it is a prime.)



              Then
              $$
              begin{aligned}
              -1 &= 1cdot 2cdot 3cdot 4cdot 5cdot 6cdot dots cdot (p-2)cdot(p-1)
              \
              &=1color{red}{cdot (-2)}cdot 3color{red}{cdot (-4)}cdot 5color{red}{cdot (-6)}cdot dots cdot (p-2)color{red}{cdot(-(p-1))}
              color{blue}{cdot(-1)^{(p-1)/2}}
              \
              &=1cdot 3cdot 5cdot dots cdot (p-2)
              color{red}{cdot (-2)cdot (-4)cdot (-6)cdot dots cdot(-(p-1))}
              color{blue}{cdot(-1)^{(p-1)/2}}
              \
              &=1cdot 3cdot 5cdot dots cdot (p-2)
              color{red}{cdot (p-2)cdot (p-4)cdot (p-6)cdot dots cdot(p-(p-1))}
              color{blue}{cdot(-1)^{(p-1)/2}}
              \
              &=(1cdot 3cdot 5cdot dots cdot (p-2))^2
              color{blue}{cdot(-1)^{(p-1)/2}}
              .
              end{aligned}
              $$
              This leads to the answer...






              share|cite|improve this answer
























                3












                3








                3






                We work in the field $Bbb F_p$ with $p$ elements. (Same as $Bbb Z/p$, for the given prime $p$, i suppose it is a prime.)



                Then
                $$
                begin{aligned}
                -1 &= 1cdot 2cdot 3cdot 4cdot 5cdot 6cdot dots cdot (p-2)cdot(p-1)
                \
                &=1color{red}{cdot (-2)}cdot 3color{red}{cdot (-4)}cdot 5color{red}{cdot (-6)}cdot dots cdot (p-2)color{red}{cdot(-(p-1))}
                color{blue}{cdot(-1)^{(p-1)/2}}
                \
                &=1cdot 3cdot 5cdot dots cdot (p-2)
                color{red}{cdot (-2)cdot (-4)cdot (-6)cdot dots cdot(-(p-1))}
                color{blue}{cdot(-1)^{(p-1)/2}}
                \
                &=1cdot 3cdot 5cdot dots cdot (p-2)
                color{red}{cdot (p-2)cdot (p-4)cdot (p-6)cdot dots cdot(p-(p-1))}
                color{blue}{cdot(-1)^{(p-1)/2}}
                \
                &=(1cdot 3cdot 5cdot dots cdot (p-2))^2
                color{blue}{cdot(-1)^{(p-1)/2}}
                .
                end{aligned}
                $$
                This leads to the answer...






                share|cite|improve this answer












                We work in the field $Bbb F_p$ with $p$ elements. (Same as $Bbb Z/p$, for the given prime $p$, i suppose it is a prime.)



                Then
                $$
                begin{aligned}
                -1 &= 1cdot 2cdot 3cdot 4cdot 5cdot 6cdot dots cdot (p-2)cdot(p-1)
                \
                &=1color{red}{cdot (-2)}cdot 3color{red}{cdot (-4)}cdot 5color{red}{cdot (-6)}cdot dots cdot (p-2)color{red}{cdot(-(p-1))}
                color{blue}{cdot(-1)^{(p-1)/2}}
                \
                &=1cdot 3cdot 5cdot dots cdot (p-2)
                color{red}{cdot (-2)cdot (-4)cdot (-6)cdot dots cdot(-(p-1))}
                color{blue}{cdot(-1)^{(p-1)/2}}
                \
                &=1cdot 3cdot 5cdot dots cdot (p-2)
                color{red}{cdot (p-2)cdot (p-4)cdot (p-6)cdot dots cdot(p-(p-1))}
                color{blue}{cdot(-1)^{(p-1)/2}}
                \
                &=(1cdot 3cdot 5cdot dots cdot (p-2))^2
                color{blue}{cdot(-1)^{(p-1)/2}}
                .
                end{aligned}
                $$
                This leads to the answer...







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 25 at 20:44









                dan_fulea

                6,2301312




                6,2301312






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • 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.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • 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.


                    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%2fmath.stackexchange.com%2fquestions%2f3013352%2fprove-that-12-cdot-32-cdot-52-cdots-p-22-equiv-1-fracp12-pm%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

                    Probability when a professor distributes a quiz and homework assignment to a class of n students.

                    Image
                    0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
                    Read more

                    Aardman Animations

                    Image
                    Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
                    Read more

                    Are they similar matrix

                    Image
                    0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
                    Read more