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

Overriding an object in memory with placement new

What does the word "veer" mean here?

How to call a function with default parameter through a pointer to function that is the return of another function?

Echoing a tail command produces unexpected output?

How do I stop a creek from eroding my steep embankment?

prime numbers and expressing non-prime numbers

What is a non-alternating simple group with big order, but relatively few conjugacy classes?

The logistics of corpse disposal

Why are there no cargo aircraft with "flying wing" design?

Is it true that "carbohydrates are of no use for the basal metabolic need"?

Using et al. for a last / senior author rather than for a first author

Why did the IBM 650 use bi-quinary?

Why do people hide their license plates in the EU?

What is Arya's weapon design?

Do I really need recursive chmod to restrict access to a folder?

Why is my conclusion inconsistent with the van't Hoff equation?

porting install scripts : can rpm replace apt?

Why is "Consequences inflicted." not a sentence?

Single word antonym of "flightless"

English words in a non-english sci-fi novel

What LEGO pieces have "real-world" functionality?

Extract all GPU name, model and GPU ram

How to tell that you are a giant?

If a contract sometimes uses the wrong name, is it still valid?



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

      7,8931320




      7,8931320












      • $begingroup$
        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

                      Can't compile dgruyter and caption packagesLaTeX templates/packages for writing a patent specificationLatex...

                      Image
                      Why does Arabsat 6A need a Falcon Heavy to launch Etiquette around loan refinance - decision is going to cost first broker a lot of money What is the PIE reconstruction for word-initial alpha with rough breathing? Is "remove commented out code" correct English? Why do I get two different answers for this counting problem? Blender 2.8 I can't see vertices, edges or faces in edit mode Where does SFDX store details about scratch orgs? 1960's book about a plague that kills all white people How can saying a song's name be a copyright violation? UK: Is there precedent for the governments e-petition site changing the
                      Read more

                      Schneeberg (Smreczany) Bibliografia | Menu...

                      Image
                      Szczyty w Niemczech paśmieSmreczanyn.p.m.Ochrza Schneeberg Państwo   Niemcy Położenie Bawaria Pasmo Fichtelgebirge Wysokość 1 051 m n.p.m. Położenie na mapie Bawarii Schneeberg Położenie na mapie Niemiec Schneeberg 50°03′N   11°51′E / 50,050000   11,850000 Multimedia w Wikimedia Commons Schneeberg – szczyt w paśmie Smreczany o wysokości 1051 m n.p.m., na zboczach którego znajdują się źródła rzeki Ochrza. Bibliografia | Rainer H. Schmeisser: Der Schneeberg, Beiträge zur Geschichts- und Landeskunde des Fichtelgebirges Nr. 1, Regensburg 1979 This page is only for reference, If you need detailed information, please check here
                      Read more

                      Hans Bellmer Spis treści Życiorys | Upamiętnienie | Przypisy | Bibliografia | Linki zewnętrzne |...

                      Image
                      Ludzie związani z III Liceum Ogólnokształcącym im. Adama Mickiewicza w KatowicachNiemieccy rzeźbiarzeNiemieccy malarze współcześniNiemieccy graficyLudzie urodzeni w KatowicachUrodzeni w 1902Zmarli w 1975 13 marca1902Katowicach23 lutego1975Paryżumalarzrzeźbiarzgrafikul. A. Mickiewiczaul. T. KościuszkiWyższej Szkole TechnicznejinstalacjiPaula EluardaBerlinie„tableaux vivants”NiemczechnazistówzdegenerowanąBerlinsurrealistówII wojnie światowejCamp des MillesAix-en-ProvenceParyżuUnicę ZürnPère-Lachaiseul. T. KościuszkiDział Grafiki im. Pawła Stellera Muzeum Historii Katowic2008Muzeum Historii KatowicNatalia KruszynaBerlinie Tablica pamiątkowa z 2008 roku na domu, w którym mieszkał artysta przy ul. T. Kościuszki 47 w Katowicach (2017) Grób Bellmera na cmentarzu Père-Lachaise Tablica pamiątkowa przy Ehrenfelsstraße 8a w Berlinie Budynek, w którym mieszkał artysta przy ul. T. Kościuszki 47 w Katowicach (2017) Hans Bellmer (ur. 13 marca 1902 w Katowicach [1] , zm. 23 lut
                      Read more