prime numbers and expressing non-prime numbers Announcing the arrival of Valued Associate...

Sci-Fi book where patients in a coma ward all live in a subconscious world linked together

List *all* the tuples!

Why am I getting the error "non-boolean type specified in a context where a condition is expected" for this request?

prime numbers and expressing non-prime numbers

Output the ŋarâþ crîþ alphabet song without using (m)any letters

Coloring maths inside a tcolorbox

How to find all the available tools in mac terminal?

What is the role of the transistor and diode in a soft start circuit?

Why is "Consequences inflicted." not a sentence?

Short Story with Cinderella as a Voo-doo Witch

Why did the IBM 650 use bi-quinary?

How to align text above triangle figure

List of Python versions

What does an IRS interview request entail when called in to verify expenses for a sole proprietor small business?

What does the "x" in "x86" represent?

What would be the ideal power source for a cybernetic eye?

51k Euros annually for a family of 4 in Berlin: Is it enough?

Why are Kinder Surprise Eggs illegal in the USA?

Extract all GPU name, model and GPU ram

How come Sam didn't become Lord of Horn Hill?

Generate an RGB colour grid

What's the purpose of writing one's academic biography in the third person?

At the end of Thor: Ragnarok why don't the Asgardians turn and head for the Bifrost as per their original plan?

Why was the term "discrete" used in discrete logarithm?



prime numbers and expressing non-prime numbers



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Expressing a Non Negative Integer as Sums of Two SquaresAre all prime numbers finite?How can can you write a prime number as a product of prime numbers?Who generates the prime numbers for encryption?In Fermat's little theorem, if mod is not prime?Prime numbers like 113Non-unique prime factorisationWhy are all non-prime numbers divisible by a prime number?Generating Prime Numbers From Composite NumbersAlternate definition of prime numbers












2












$begingroup$


My textbook says if $b$ is a non-prime number then it can be expressed as a product of prime numbers. But if $1$ isn't prime how it can be expressed as a product of prime numbers?






number-theory elementary-number-theory prime-numbers







share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    An empty product is still a product.
    $endgroup$
    – lulu
    4 hours ago






  • 1




    $begingroup$
    The standard modern day view is that the number 1 is neither prime nor composite.
    $endgroup$
    – Martin Hansen
    4 hours ago
















2












$begingroup$


My textbook says if $b$ is a non-prime number then it can be expressed as a product of prime numbers. But if $1$ isn't prime how it can be expressed as a product of prime numbers?






number-theory elementary-number-theory prime-numbers







share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    An empty product is still a product.
    $endgroup$
    – lulu
    4 hours ago






  • 1




    $begingroup$
    The standard modern day view is that the number 1 is neither prime nor composite.
    $endgroup$
    – Martin Hansen
    4 hours ago














2












2








2





$begingroup$


My textbook says if $b$ is a non-prime number then it can be expressed as a product of prime numbers. But if $1$ isn't prime how it can be expressed as a product of prime numbers?






number-theory elementary-number-theory prime-numbers







share|cite|improve this question











$endgroup$




My textbook says if $b$ is a non-prime number then it can be expressed as a product of prime numbers. But if $1$ isn't prime how it can be expressed as a product of prime numbers?






number-theory elementary-number-theory prime-numbers




number-theory elementary-number-theory prime-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago









Mr. Brooks

22211338




22211338










asked 4 hours ago









Ahmed M. ElsonbatyAhmed M. Elsonbaty

624




624








  • 5




    $begingroup$
    An empty product is still a product.
    $endgroup$
    – lulu
    4 hours ago






  • 1




    $begingroup$
    The standard modern day view is that the number 1 is neither prime nor composite.
    $endgroup$
    – Martin Hansen
    4 hours ago














  • 5




    $begingroup$
    An empty product is still a product.
    $endgroup$
    – lulu
    4 hours ago






  • 1




    $begingroup$
    The standard modern day view is that the number 1 is neither prime nor composite.
    $endgroup$
    – Martin Hansen
    4 hours ago








5




5




$begingroup$
An empty product is still a product.
$endgroup$
– lulu
4 hours ago




$begingroup$
An empty product is still a product.
$endgroup$
– lulu
4 hours ago




1




1




$begingroup$
The standard modern day view is that the number 1 is neither prime nor composite.
$endgroup$
– Martin Hansen
4 hours ago




$begingroup$
The standard modern day view is that the number 1 is neither prime nor composite.
$endgroup$
– Martin Hansen
4 hours ago










3 Answers
3






active

oldest

votes


















1












$begingroup$

"In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors."



Your error lies in stating the fundamental theorem of arithmetic incorrectly.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    What about non-UFDs? 1 is still not prime in those, so FTA does not apply, and the asker didn't even bring up the FTA.
    $endgroup$
    – Mr. Brooks
    3 hours ago










  • $begingroup$
    I assumed that their textbook meant the FTA and they accepted my answer, what is the problem?
    $endgroup$
    – Peter Foreman
    3 hours ago






  • 1




    $begingroup$
    There is no problem at all if you treat this website as a competition for points.
    $endgroup$
    – Mr. Brooks
    3 hours ago



















2












$begingroup$

This is mainly just an extended comment on Peter Foreman's answer. The (relatively difficult) uniqueness aspect of the Fundamental Theorem of Arithmetic is not needed for the OP's question, just the (easier) existence aspect.



What's missing from the OP's textbook is the qualifier in the correct assertion that every non-prime number greater than $1$ can be expressed as a product of primes. This is the existence aspect of FTA, and it can be proved by strong induction: If $ngt1$ is not a prime, then $n=ab$ for some pair of integers with $1lt a,b$. Both $a$ and $b$ must be less than $n$ (otherwise their product would be more than $n$), so we can assume, by strong induction, that each of them can be written as a product of primes, hence so can their product, which is $n$.



Remark: "Strong" induction means that you don't just assume an assertion is true for $n-1$ and then prove it for $n$, you assume it's true for all positive integers $klt n$. In this case the assertion is "if $kgt1$ and $k$ is non-prime, then $k$ can be written as a product of primes." Note that the base case, $k=1$, is vacuously true, because $1$ is not greater than $1$.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    What is the sum of no numbers at all? Zero, of course, since it is the additive identity: $x + 0 = 0$, where $x neq 0$, or even if it is.



    Now, what is the product of no numbers at all? It can't be zero, since, maintaining the stipulation that $x neq 0$, we have $x times 0 = 0$, and we said $x neq 0$. The multiplicative identity is $1$, since $x times 1 = 1$.



    Hence, the product of no primes at all is $1$. The fundamental theorem of arithmetic is a subtlety that's unnecessary for answering your question.






    share|cite|improve this answer









    $endgroup$














      Your Answer








      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%2f3190287%2fprime-numbers-and-expressing-non-prime-numbers%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1












      $begingroup$

      "In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors."



      Your error lies in stating the fundamental theorem of arithmetic incorrectly.






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        What about non-UFDs? 1 is still not prime in those, so FTA does not apply, and the asker didn't even bring up the FTA.
        $endgroup$
        – Mr. Brooks
        3 hours ago










      • $begingroup$
        I assumed that their textbook meant the FTA and they accepted my answer, what is the problem?
        $endgroup$
        – Peter Foreman
        3 hours ago






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        3 hours ago
















      1












      $begingroup$

      "In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors."



      Your error lies in stating the fundamental theorem of arithmetic incorrectly.






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        What about non-UFDs? 1 is still not prime in those, so FTA does not apply, and the asker didn't even bring up the FTA.
        $endgroup$
        – Mr. Brooks
        3 hours ago










      • $begingroup$
        I assumed that their textbook meant the FTA and they accepted my answer, what is the problem?
        $endgroup$
        – Peter Foreman
        3 hours ago






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        3 hours ago














      1












      1








      1





      $begingroup$

      "In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors."



      Your error lies in stating the fundamental theorem of arithmetic incorrectly.






      share|cite|improve this answer









      $endgroup$



      "In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors."



      Your error lies in stating the fundamental theorem of arithmetic incorrectly.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered 4 hours ago









      Peter ForemanPeter Foreman

      7,8921320




      7,8921320












      • $begingroup$
        What about non-UFDs? 1 is still not prime in those, so FTA does not apply, and the asker didn't even bring up the FTA.
        $endgroup$
        – Mr. Brooks
        3 hours ago










      • $begingroup$
        I assumed that their textbook meant the FTA and they accepted my answer, what is the problem?
        $endgroup$
        – Peter Foreman
        3 hours ago






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        3 hours ago


















      • $begingroup$
        What about non-UFDs? 1 is still not prime in those, so FTA does not apply, and the asker didn't even bring up the FTA.
        $endgroup$
        – Mr. Brooks
        3 hours ago










      • $begingroup$
        I assumed that their textbook meant the FTA and they accepted my answer, what is the problem?
        $endgroup$
        – Peter Foreman
        3 hours ago






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        3 hours ago
















      $begingroup$
      What about non-UFDs? 1 is still not prime in those, so FTA does not apply, and the asker didn't even bring up the FTA.
      $endgroup$
      – Mr. Brooks
      3 hours ago




      $begingroup$
      What about non-UFDs? 1 is still not prime in those, so FTA does not apply, and the asker didn't even bring up the FTA.
      $endgroup$
      – Mr. Brooks
      3 hours ago












      $begingroup$
      I assumed that their textbook meant the FTA and they accepted my answer, what is the problem?
      $endgroup$
      – Peter Foreman
      3 hours ago




      $begingroup$
      I assumed that their textbook meant the FTA and they accepted my answer, what is the problem?
      $endgroup$
      – Peter Foreman
      3 hours ago




      1




      1




      $begingroup$
      There is no problem at all if you treat this website as a competition for points.
      $endgroup$
      – Mr. Brooks
      3 hours ago




      $begingroup$
      There is no problem at all if you treat this website as a competition for points.
      $endgroup$
      – Mr. Brooks
      3 hours ago











      2












      $begingroup$

      This is mainly just an extended comment on Peter Foreman's answer. The (relatively difficult) uniqueness aspect of the Fundamental Theorem of Arithmetic is not needed for the OP's question, just the (easier) existence aspect.



      What's missing from the OP's textbook is the qualifier in the correct assertion that every non-prime number greater than $1$ can be expressed as a product of primes. This is the existence aspect of FTA, and it can be proved by strong induction: If $ngt1$ is not a prime, then $n=ab$ for some pair of integers with $1lt a,b$. Both $a$ and $b$ must be less than $n$ (otherwise their product would be more than $n$), so we can assume, by strong induction, that each of them can be written as a product of primes, hence so can their product, which is $n$.



      Remark: "Strong" induction means that you don't just assume an assertion is true for $n-1$ and then prove it for $n$, you assume it's true for all positive integers $klt n$. In this case the assertion is "if $kgt1$ and $k$ is non-prime, then $k$ can be written as a product of primes." Note that the base case, $k=1$, is vacuously true, because $1$ is not greater than $1$.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        This is mainly just an extended comment on Peter Foreman's answer. The (relatively difficult) uniqueness aspect of the Fundamental Theorem of Arithmetic is not needed for the OP's question, just the (easier) existence aspect.



        What's missing from the OP's textbook is the qualifier in the correct assertion that every non-prime number greater than $1$ can be expressed as a product of primes. This is the existence aspect of FTA, and it can be proved by strong induction: If $ngt1$ is not a prime, then $n=ab$ for some pair of integers with $1lt a,b$. Both $a$ and $b$ must be less than $n$ (otherwise their product would be more than $n$), so we can assume, by strong induction, that each of them can be written as a product of primes, hence so can their product, which is $n$.



        Remark: "Strong" induction means that you don't just assume an assertion is true for $n-1$ and then prove it for $n$, you assume it's true for all positive integers $klt n$. In this case the assertion is "if $kgt1$ and $k$ is non-prime, then $k$ can be written as a product of primes." Note that the base case, $k=1$, is vacuously true, because $1$ is not greater than $1$.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          This is mainly just an extended comment on Peter Foreman's answer. The (relatively difficult) uniqueness aspect of the Fundamental Theorem of Arithmetic is not needed for the OP's question, just the (easier) existence aspect.



          What's missing from the OP's textbook is the qualifier in the correct assertion that every non-prime number greater than $1$ can be expressed as a product of primes. This is the existence aspect of FTA, and it can be proved by strong induction: If $ngt1$ is not a prime, then $n=ab$ for some pair of integers with $1lt a,b$. Both $a$ and $b$ must be less than $n$ (otherwise their product would be more than $n$), so we can assume, by strong induction, that each of them can be written as a product of primes, hence so can their product, which is $n$.



          Remark: "Strong" induction means that you don't just assume an assertion is true for $n-1$ and then prove it for $n$, you assume it's true for all positive integers $klt n$. In this case the assertion is "if $kgt1$ and $k$ is non-prime, then $k$ can be written as a product of primes." Note that the base case, $k=1$, is vacuously true, because $1$ is not greater than $1$.






          share|cite|improve this answer









          $endgroup$



          This is mainly just an extended comment on Peter Foreman's answer. The (relatively difficult) uniqueness aspect of the Fundamental Theorem of Arithmetic is not needed for the OP's question, just the (easier) existence aspect.



          What's missing from the OP's textbook is the qualifier in the correct assertion that every non-prime number greater than $1$ can be expressed as a product of primes. This is the existence aspect of FTA, and it can be proved by strong induction: If $ngt1$ is not a prime, then $n=ab$ for some pair of integers with $1lt a,b$. Both $a$ and $b$ must be less than $n$ (otherwise their product would be more than $n$), so we can assume, by strong induction, that each of them can be written as a product of primes, hence so can their product, which is $n$.



          Remark: "Strong" induction means that you don't just assume an assertion is true for $n-1$ and then prove it for $n$, you assume it's true for all positive integers $klt n$. In this case the assertion is "if $kgt1$ and $k$ is non-prime, then $k$ can be written as a product of primes." Note that the base case, $k=1$, is vacuously true, because $1$ is not greater than $1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Barry CipraBarry Cipra

          60.7k655129




          60.7k655129























              1












              $begingroup$

              What is the sum of no numbers at all? Zero, of course, since it is the additive identity: $x + 0 = 0$, where $x neq 0$, or even if it is.



              Now, what is the product of no numbers at all? It can't be zero, since, maintaining the stipulation that $x neq 0$, we have $x times 0 = 0$, and we said $x neq 0$. The multiplicative identity is $1$, since $x times 1 = 1$.



              Hence, the product of no primes at all is $1$. The fundamental theorem of arithmetic is a subtlety that's unnecessary for answering your question.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                What is the sum of no numbers at all? Zero, of course, since it is the additive identity: $x + 0 = 0$, where $x neq 0$, or even if it is.



                Now, what is the product of no numbers at all? It can't be zero, since, maintaining the stipulation that $x neq 0$, we have $x times 0 = 0$, and we said $x neq 0$. The multiplicative identity is $1$, since $x times 1 = 1$.



                Hence, the product of no primes at all is $1$. The fundamental theorem of arithmetic is a subtlety that's unnecessary for answering your question.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  What is the sum of no numbers at all? Zero, of course, since it is the additive identity: $x + 0 = 0$, where $x neq 0$, or even if it is.



                  Now, what is the product of no numbers at all? It can't be zero, since, maintaining the stipulation that $x neq 0$, we have $x times 0 = 0$, and we said $x neq 0$. The multiplicative identity is $1$, since $x times 1 = 1$.



                  Hence, the product of no primes at all is $1$. The fundamental theorem of arithmetic is a subtlety that's unnecessary for answering your question.






                  share|cite|improve this answer









                  $endgroup$



                  What is the sum of no numbers at all? Zero, of course, since it is the additive identity: $x + 0 = 0$, where $x neq 0$, or even if it is.



                  Now, what is the product of no numbers at all? It can't be zero, since, maintaining the stipulation that $x neq 0$, we have $x times 0 = 0$, and we said $x neq 0$. The multiplicative identity is $1$, since $x times 1 = 1$.



                  Hence, the product of no primes at all is $1$. The fundamental theorem of arithmetic is a subtlety that's unnecessary for answering your question.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  Mr. BrooksMr. Brooks

                  22211338




                  22211338






























                      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.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3190287%2fprime-numbers-and-expressing-non-prime-numbers%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

                      Why do type traits not work with types in namespace scope?What are POD types in C++?Why can templates only be...

                      Image
                      What does it mean to express a gate in Dirac notation? A Strange Latex Symbol How can Republicans who favour free markets, consistently express anger when they don't like the outcome of that choice? What happened to Captain America in Endgame? How to solve constants out of the internal energy equation? How can I place the product on a social media post better? Will tsunami waves travel forever if there was no land? Unexpected email from Yorkshire Bank What is the relationship between spectral sequences and obstruction theory? How can I practically buy stocks? Will a top journal at least re
                      Read more

                      Will tsunami waves travel forever if there was no land?Why do tsunami waves begin with the water flowing away...

                      Image
                      How much cash can I safely carry into the USA and avoid civil forfeiture? Is there really no use for MD5 anymore? Is contacting this expert in the field something acceptable or would it be discourteous? How to pronounce 'C++' in Spanish How to creep the reader out with what seems like a normal person? Critique of timeline aesthetic Sci fi novel series with instant travel between planets through gates. A river runs through the gates How did Captain America manage to do this? Are Boeing 737-800’s grounded? Error message with tabularx How to make a pipeline wait for end-of-file or stop af
                      Read more

                      Should I use Docker or LXD?How to cache (more) data on SSD/RAM to avoid spin up?Unable to get Windows File...

                      Image
                      kill -0 <PID> は何をするのでしょうか？ Why avoid shared user accounts? Using only 1s, make 29 with the minimum number of digits Cookies - Should the toggles be on? Can I string the DnD Starter Set campaign into another modules, keeping the same characters? Why do neural networks need so many training examples to perform? Intern applicant asking for compensation equivalent to that of permanent employee Why would space fleets be aligned? It took me a lot of time to make this, pls like. (YouTube Comments #1) Can a long polymer chain interact with itself via van der Waals forces? What is the purpose of
                      Read more