prime numbers and expressing non-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
    6 hours ago






  • 1




    $begingroup$
    The standard modern day view is that the number 1 is neither prime nor composite.
    $endgroup$
    – Martin Hansen
    6 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
    6 hours ago






  • 1




    $begingroup$
    The standard modern day view is that the number 1 is neither prime nor composite.
    $endgroup$
    – Martin Hansen
    6 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 5 hours ago









Mr. Brooks

22211338




22211338










asked 6 hours ago









Ahmed M. ElsonbatyAhmed M. Elsonbaty

624




624








  • 5




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






  • 1




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














  • 5




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






  • 1




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








5




5




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




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




1




1




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




$begingroup$
The standard modern day view is that the number 1 is neither prime nor composite.
$endgroup$
– Martin Hansen
6 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
    5 hours ago










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






  • 1




    $begingroup$
    There is no problem at all if you treat this website as a competition for points.
    $endgroup$
    – Mr. Brooks
    5 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









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      3 Answers
      3






      active

      oldest

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      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
        5 hours ago










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






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        5 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
        5 hours ago










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






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        5 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 6 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
        5 hours ago










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






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        5 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
        5 hours ago










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






      • 1




        $begingroup$
        There is no problem at all if you treat this website as a competition for points.
        $endgroup$
        – Mr. Brooks
        5 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
      5 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
      5 hours ago












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




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




      1




      1




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




      $begingroup$
      There is no problem at all if you treat this website as a competition for points.
      $endgroup$
      – Mr. Brooks
      5 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 5 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 5 hours ago









                  Mr. BrooksMr. Brooks

                  22211338




                  22211338






























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                      Tu-95轟炸機

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                      body.skin-minerva .mw-parser-output table.infobox caption{text-align:center} 图-95 Ту–95 北约代号：熊（Bear） 一架在安格斯空军基地的图-95MS 概觀 類型 战略轰炸机 代號 北约代号： Bear （熊） 乘員 7名 駕駛員2名，機尾炮手1名，其他人員4名 首飛 1952年11月12日 服役 1956年 設計 圖波列夫設計局 產量 超过500架 現況 现役 主要用戶   蘇聯   俄羅斯 衍生機型 图-114、图-119、Tu-95 技术数据 長度 49.50米（162呎5吋） 翼展 51.10米（167呎8吋） 高度 12.12米（39呎9吋） 翼面積 310平方米（3,330平方呎） 空重 90,000公斤（198,000磅） 最大起飛重量 188,000公斤（414,500磅） 發動機 4具库兹涅佐夫NK-12MV型渦輪螺旋槳發動機 功率 4×11,000千瓦（14,800馬力） 性能數據 最大速度 925公里／時（500節，575哩／時） 爬升率 10米／秒（2,000呎／分） 最大升限 12,000米（39,000呎） 最大航程 15,000公里（8,100海浬，9,400哩） 翼負荷 606公斤／平方米（124磅／平方呎） 推重比 235 W／公斤（0.143馬力／磅） 武器装备 機炮 1具/2具AM-23型23公釐雷達控制機砲（機尾） 飛彈 空对地导弹： Kh-20、Kh-22 Kh-26、Kh-55 炸彈 60枚FAB-250炸彈，或 30枚FAB-500炸彈。 其他 载弹量（包括飞弹）：最多15,000公斤（33,000磅） Tu-95熊式MR型機 图-95 （俄语： Ту–95 ， 英语： Tu-95 ），北約代號： Bear （ 熊 ），是蘇聯圖波列夫設計局所研製，是全世界唯一服役的大型四渦輪螺旋槳發動機...
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