Find all natural numbers $n$, such that polynomial $n^7+n^6+n^5+1$ would have exactly 3 divisors.












3












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Find all natural numbers $n$, such that polynomial $$n^7+n^6+n^5+1$$ would have exactly 3 divisors.




What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors. Also, n must be an even number, otherwise this polynomial would be divisible by 4 (if it's an odd number, then it would be possible and would fit conditions only and only if n=1). My guts are telling me that somehow I have to prove that it's impossible with even numbers too, though I have no idea how.










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    3












    $begingroup$



    Find all natural numbers $n$, such that polynomial $$n^7+n^6+n^5+1$$ would have exactly 3 divisors.




    What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors. Also, n must be an even number, otherwise this polynomial would be divisible by 4 (if it's an odd number, then it would be possible and would fit conditions only and only if n=1). My guts are telling me that somehow I have to prove that it's impossible with even numbers too, though I have no idea how.










    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$



      Find all natural numbers $n$, such that polynomial $$n^7+n^6+n^5+1$$ would have exactly 3 divisors.




      What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors. Also, n must be an even number, otherwise this polynomial would be divisible by 4 (if it's an odd number, then it would be possible and would fit conditions only and only if n=1). My guts are telling me that somehow I have to prove that it's impossible with even numbers too, though I have no idea how.










      share|cite|improve this question











      $endgroup$





      Find all natural numbers $n$, such that polynomial $$n^7+n^6+n^5+1$$ would have exactly 3 divisors.




      What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors. Also, n must be an even number, otherwise this polynomial would be divisible by 4 (if it's an odd number, then it would be possible and would fit conditions only and only if n=1). My guts are telling me that somehow I have to prove that it's impossible with even numbers too, though I have no idea how.







      number-theory elementary-number-theory polynomials






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      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 2 '18 at 19:32









      greedoid

      38.7k114797




      38.7k114797










      asked Dec 2 '18 at 18:36









      Severus156Severus156

      1637




      1637






















          2 Answers
          2






          active

          oldest

          votes


















          6












          $begingroup$

          You can factorize this polynomial like this: $$(n+1)(n^2+1)(n^4-n+1)$$ so this one has almost always at least 3 divisors.



          Actualy, for $n>1$ we have $n+1>2$, $n^2+1>2$ and $n^4-n+1>2$ so it could be only $n=1$ which works.





          How I got this this factorization:
          begin{eqnarray} n^7+n^6+n^5+1 &=& n^5(n^2+1) + (n^2+1)(n^4-n^2+1)\
          &=& (n^2+1)(n^5+n^4-n^2+1)\
          &=& (n^2+1)Big(n^4(n+1)-(n-1)(n+1)Big)\
          &=&(n+1)(n^2+1)(n^4-n+1)
          end{eqnarray}






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Seems approvable. And also I'm quite interested - how do you create such hard factorizations for such untraditional polynomials? Is there any secret or are you just using some programs/apps for this?
            $endgroup$
            – Severus156
            Dec 2 '18 at 18:49






          • 1




            $begingroup$
            @Severus156 You sometimes see a root easily; like in this case: -1 is an easy to find root. So you can divide the polynomial by $n+1$. Then repeat the process.
            $endgroup$
            – Guus Palmer
            Dec 2 '18 at 18:51








          • 1




            $begingroup$
            But further roots are not as beautiful as before, $n^2+1$ root is $i$, so basically you just guess every single integer close to $0$ and hope that you will find those roots, is that so @GuusPalmer?
            $endgroup$
            – Severus156
            Dec 2 '18 at 18:54








          • 1




            $begingroup$
            @Severus156 That's what I would do most of the time. But you should also be aware that Number Theory is full of tricks; i.e. there's not always a general recipe for everything, just look at Mordell equations: they use a large variety of methods to solve these. To not wander away from the subject too much, notice that the polynomial has an even degree and two unevens in it. That sometimes indicates a $pm1$ or $pm i$...
            $endgroup$
            – Guus Palmer
            Dec 2 '18 at 19:00












          • $begingroup$
            "Actualy, for n>1 we have n+1>2, n2+1>2 and n4−n+1>2 so it could be only n=1 which works" Why is that relevant? We will have exactly 3 factors if all $n+1$ and $n^2 + 1$ and $n^4 - n + 1$ are prime.. Why can't that happen if $n > 1$?
            $endgroup$
            – fleablood
            Dec 2 '18 at 20:26





















          0












          $begingroup$


          What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors.




          Or fewer.



          To rephrase and elaborate on greedoid's answer in a way that doesn't involve complete factoring:



          So if if the polynomial factor as $r(n)q(n)$ we must have either $r(n) = 1; q(n) = p^2$ for some $n$ and prime $p$ or $r(n) =q(n)=p$ for some $n$ and prime $p$.



          Note that if $n =-1$ we get the polynomial is $0$ so $n+1$ factors.



          And $frac {n^7 + n^6 + n^5 + 1}{n+1} = n^6 + n^4 -n^3 +n^2 -n + 1$.



          If $n+1 = 1$ then we get $n = 0$ and .... the other is $1$ and that won't do.



          It's pretty clear if $n > 1$ then $n^6 + ... etc > 1$ so $(n+1) = p^2$ and $n^6 + ... etc = 1$ is out of the question.



          That just leaves $n + 1 = n^6 + .... = p$ as an option. And for that to work we must have $n = 1$. And in that case it does work out that $n+1 = n^6 + ... etc = 2$.






          share|cite|improve this answer









          $endgroup$













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            2 Answers
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            6












            $begingroup$

            You can factorize this polynomial like this: $$(n+1)(n^2+1)(n^4-n+1)$$ so this one has almost always at least 3 divisors.



            Actualy, for $n>1$ we have $n+1>2$, $n^2+1>2$ and $n^4-n+1>2$ so it could be only $n=1$ which works.





            How I got this this factorization:
            begin{eqnarray} n^7+n^6+n^5+1 &=& n^5(n^2+1) + (n^2+1)(n^4-n^2+1)\
            &=& (n^2+1)(n^5+n^4-n^2+1)\
            &=& (n^2+1)Big(n^4(n+1)-(n-1)(n+1)Big)\
            &=&(n+1)(n^2+1)(n^4-n+1)
            end{eqnarray}






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Seems approvable. And also I'm quite interested - how do you create such hard factorizations for such untraditional polynomials? Is there any secret or are you just using some programs/apps for this?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:49






            • 1




              $begingroup$
              @Severus156 You sometimes see a root easily; like in this case: -1 is an easy to find root. So you can divide the polynomial by $n+1$. Then repeat the process.
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 18:51








            • 1




              $begingroup$
              But further roots are not as beautiful as before, $n^2+1$ root is $i$, so basically you just guess every single integer close to $0$ and hope that you will find those roots, is that so @GuusPalmer?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:54








            • 1




              $begingroup$
              @Severus156 That's what I would do most of the time. But you should also be aware that Number Theory is full of tricks; i.e. there's not always a general recipe for everything, just look at Mordell equations: they use a large variety of methods to solve these. To not wander away from the subject too much, notice that the polynomial has an even degree and two unevens in it. That sometimes indicates a $pm1$ or $pm i$...
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 19:00












            • $begingroup$
              "Actualy, for n>1 we have n+1>2, n2+1>2 and n4−n+1>2 so it could be only n=1 which works" Why is that relevant? We will have exactly 3 factors if all $n+1$ and $n^2 + 1$ and $n^4 - n + 1$ are prime.. Why can't that happen if $n > 1$?
              $endgroup$
              – fleablood
              Dec 2 '18 at 20:26


















            6












            $begingroup$

            You can factorize this polynomial like this: $$(n+1)(n^2+1)(n^4-n+1)$$ so this one has almost always at least 3 divisors.



            Actualy, for $n>1$ we have $n+1>2$, $n^2+1>2$ and $n^4-n+1>2$ so it could be only $n=1$ which works.





            How I got this this factorization:
            begin{eqnarray} n^7+n^6+n^5+1 &=& n^5(n^2+1) + (n^2+1)(n^4-n^2+1)\
            &=& (n^2+1)(n^5+n^4-n^2+1)\
            &=& (n^2+1)Big(n^4(n+1)-(n-1)(n+1)Big)\
            &=&(n+1)(n^2+1)(n^4-n+1)
            end{eqnarray}






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Seems approvable. And also I'm quite interested - how do you create such hard factorizations for such untraditional polynomials? Is there any secret or are you just using some programs/apps for this?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:49






            • 1




              $begingroup$
              @Severus156 You sometimes see a root easily; like in this case: -1 is an easy to find root. So you can divide the polynomial by $n+1$. Then repeat the process.
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 18:51








            • 1




              $begingroup$
              But further roots are not as beautiful as before, $n^2+1$ root is $i$, so basically you just guess every single integer close to $0$ and hope that you will find those roots, is that so @GuusPalmer?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:54








            • 1




              $begingroup$
              @Severus156 That's what I would do most of the time. But you should also be aware that Number Theory is full of tricks; i.e. there's not always a general recipe for everything, just look at Mordell equations: they use a large variety of methods to solve these. To not wander away from the subject too much, notice that the polynomial has an even degree and two unevens in it. That sometimes indicates a $pm1$ or $pm i$...
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 19:00












            • $begingroup$
              "Actualy, for n>1 we have n+1>2, n2+1>2 and n4−n+1>2 so it could be only n=1 which works" Why is that relevant? We will have exactly 3 factors if all $n+1$ and $n^2 + 1$ and $n^4 - n + 1$ are prime.. Why can't that happen if $n > 1$?
              $endgroup$
              – fleablood
              Dec 2 '18 at 20:26
















            6












            6








            6





            $begingroup$

            You can factorize this polynomial like this: $$(n+1)(n^2+1)(n^4-n+1)$$ so this one has almost always at least 3 divisors.



            Actualy, for $n>1$ we have $n+1>2$, $n^2+1>2$ and $n^4-n+1>2$ so it could be only $n=1$ which works.





            How I got this this factorization:
            begin{eqnarray} n^7+n^6+n^5+1 &=& n^5(n^2+1) + (n^2+1)(n^4-n^2+1)\
            &=& (n^2+1)(n^5+n^4-n^2+1)\
            &=& (n^2+1)Big(n^4(n+1)-(n-1)(n+1)Big)\
            &=&(n+1)(n^2+1)(n^4-n+1)
            end{eqnarray}






            share|cite|improve this answer











            $endgroup$



            You can factorize this polynomial like this: $$(n+1)(n^2+1)(n^4-n+1)$$ so this one has almost always at least 3 divisors.



            Actualy, for $n>1$ we have $n+1>2$, $n^2+1>2$ and $n^4-n+1>2$ so it could be only $n=1$ which works.





            How I got this this factorization:
            begin{eqnarray} n^7+n^6+n^5+1 &=& n^5(n^2+1) + (n^2+1)(n^4-n^2+1)\
            &=& (n^2+1)(n^5+n^4-n^2+1)\
            &=& (n^2+1)Big(n^4(n+1)-(n-1)(n+1)Big)\
            &=&(n+1)(n^2+1)(n^4-n+1)
            end{eqnarray}







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 2 '18 at 18:52

























            answered Dec 2 '18 at 18:41









            greedoidgreedoid

            38.7k114797




            38.7k114797












            • $begingroup$
              Seems approvable. And also I'm quite interested - how do you create such hard factorizations for such untraditional polynomials? Is there any secret or are you just using some programs/apps for this?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:49






            • 1




              $begingroup$
              @Severus156 You sometimes see a root easily; like in this case: -1 is an easy to find root. So you can divide the polynomial by $n+1$. Then repeat the process.
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 18:51








            • 1




              $begingroup$
              But further roots are not as beautiful as before, $n^2+1$ root is $i$, so basically you just guess every single integer close to $0$ and hope that you will find those roots, is that so @GuusPalmer?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:54








            • 1




              $begingroup$
              @Severus156 That's what I would do most of the time. But you should also be aware that Number Theory is full of tricks; i.e. there's not always a general recipe for everything, just look at Mordell equations: they use a large variety of methods to solve these. To not wander away from the subject too much, notice that the polynomial has an even degree and two unevens in it. That sometimes indicates a $pm1$ or $pm i$...
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 19:00












            • $begingroup$
              "Actualy, for n>1 we have n+1>2, n2+1>2 and n4−n+1>2 so it could be only n=1 which works" Why is that relevant? We will have exactly 3 factors if all $n+1$ and $n^2 + 1$ and $n^4 - n + 1$ are prime.. Why can't that happen if $n > 1$?
              $endgroup$
              – fleablood
              Dec 2 '18 at 20:26




















            • $begingroup$
              Seems approvable. And also I'm quite interested - how do you create such hard factorizations for such untraditional polynomials? Is there any secret or are you just using some programs/apps for this?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:49






            • 1




              $begingroup$
              @Severus156 You sometimes see a root easily; like in this case: -1 is an easy to find root. So you can divide the polynomial by $n+1$. Then repeat the process.
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 18:51








            • 1




              $begingroup$
              But further roots are not as beautiful as before, $n^2+1$ root is $i$, so basically you just guess every single integer close to $0$ and hope that you will find those roots, is that so @GuusPalmer?
              $endgroup$
              – Severus156
              Dec 2 '18 at 18:54








            • 1




              $begingroup$
              @Severus156 That's what I would do most of the time. But you should also be aware that Number Theory is full of tricks; i.e. there's not always a general recipe for everything, just look at Mordell equations: they use a large variety of methods to solve these. To not wander away from the subject too much, notice that the polynomial has an even degree and two unevens in it. That sometimes indicates a $pm1$ or $pm i$...
              $endgroup$
              – Guus Palmer
              Dec 2 '18 at 19:00












            • $begingroup$
              "Actualy, for n>1 we have n+1>2, n2+1>2 and n4−n+1>2 so it could be only n=1 which works" Why is that relevant? We will have exactly 3 factors if all $n+1$ and $n^2 + 1$ and $n^4 - n + 1$ are prime.. Why can't that happen if $n > 1$?
              $endgroup$
              – fleablood
              Dec 2 '18 at 20:26


















            $begingroup$
            Seems approvable. And also I'm quite interested - how do you create such hard factorizations for such untraditional polynomials? Is there any secret or are you just using some programs/apps for this?
            $endgroup$
            – Severus156
            Dec 2 '18 at 18:49




            $begingroup$
            Seems approvable. And also I'm quite interested - how do you create such hard factorizations for such untraditional polynomials? Is there any secret or are you just using some programs/apps for this?
            $endgroup$
            – Severus156
            Dec 2 '18 at 18:49




            1




            1




            $begingroup$
            @Severus156 You sometimes see a root easily; like in this case: -1 is an easy to find root. So you can divide the polynomial by $n+1$. Then repeat the process.
            $endgroup$
            – Guus Palmer
            Dec 2 '18 at 18:51






            $begingroup$
            @Severus156 You sometimes see a root easily; like in this case: -1 is an easy to find root. So you can divide the polynomial by $n+1$. Then repeat the process.
            $endgroup$
            – Guus Palmer
            Dec 2 '18 at 18:51






            1




            1




            $begingroup$
            But further roots are not as beautiful as before, $n^2+1$ root is $i$, so basically you just guess every single integer close to $0$ and hope that you will find those roots, is that so @GuusPalmer?
            $endgroup$
            – Severus156
            Dec 2 '18 at 18:54






            $begingroup$
            But further roots are not as beautiful as before, $n^2+1$ root is $i$, so basically you just guess every single integer close to $0$ and hope that you will find those roots, is that so @GuusPalmer?
            $endgroup$
            – Severus156
            Dec 2 '18 at 18:54






            1




            1




            $begingroup$
            @Severus156 That's what I would do most of the time. But you should also be aware that Number Theory is full of tricks; i.e. there's not always a general recipe for everything, just look at Mordell equations: they use a large variety of methods to solve these. To not wander away from the subject too much, notice that the polynomial has an even degree and two unevens in it. That sometimes indicates a $pm1$ or $pm i$...
            $endgroup$
            – Guus Palmer
            Dec 2 '18 at 19:00






            $begingroup$
            @Severus156 That's what I would do most of the time. But you should also be aware that Number Theory is full of tricks; i.e. there's not always a general recipe for everything, just look at Mordell equations: they use a large variety of methods to solve these. To not wander away from the subject too much, notice that the polynomial has an even degree and two unevens in it. That sometimes indicates a $pm1$ or $pm i$...
            $endgroup$
            – Guus Palmer
            Dec 2 '18 at 19:00














            $begingroup$
            "Actualy, for n>1 we have n+1>2, n2+1>2 and n4−n+1>2 so it could be only n=1 which works" Why is that relevant? We will have exactly 3 factors if all $n+1$ and $n^2 + 1$ and $n^4 - n + 1$ are prime.. Why can't that happen if $n > 1$?
            $endgroup$
            – fleablood
            Dec 2 '18 at 20:26






            $begingroup$
            "Actualy, for n>1 we have n+1>2, n2+1>2 and n4−n+1>2 so it could be only n=1 which works" Why is that relevant? We will have exactly 3 factors if all $n+1$ and $n^2 + 1$ and $n^4 - n + 1$ are prime.. Why can't that happen if $n > 1$?
            $endgroup$
            – fleablood
            Dec 2 '18 at 20:26













            0












            $begingroup$


            What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors.




            Or fewer.



            To rephrase and elaborate on greedoid's answer in a way that doesn't involve complete factoring:



            So if if the polynomial factor as $r(n)q(n)$ we must have either $r(n) = 1; q(n) = p^2$ for some $n$ and prime $p$ or $r(n) =q(n)=p$ for some $n$ and prime $p$.



            Note that if $n =-1$ we get the polynomial is $0$ so $n+1$ factors.



            And $frac {n^7 + n^6 + n^5 + 1}{n+1} = n^6 + n^4 -n^3 +n^2 -n + 1$.



            If $n+1 = 1$ then we get $n = 0$ and .... the other is $1$ and that won't do.



            It's pretty clear if $n > 1$ then $n^6 + ... etc > 1$ so $(n+1) = p^2$ and $n^6 + ... etc = 1$ is out of the question.



            That just leaves $n + 1 = n^6 + .... = p$ as an option. And for that to work we must have $n = 1$. And in that case it does work out that $n+1 = n^6 + ... etc = 2$.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$


              What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors.




              Or fewer.



              To rephrase and elaborate on greedoid's answer in a way that doesn't involve complete factoring:



              So if if the polynomial factor as $r(n)q(n)$ we must have either $r(n) = 1; q(n) = p^2$ for some $n$ and prime $p$ or $r(n) =q(n)=p$ for some $n$ and prime $p$.



              Note that if $n =-1$ we get the polynomial is $0$ so $n+1$ factors.



              And $frac {n^7 + n^6 + n^5 + 1}{n+1} = n^6 + n^4 -n^3 +n^2 -n + 1$.



              If $n+1 = 1$ then we get $n = 0$ and .... the other is $1$ and that won't do.



              It's pretty clear if $n > 1$ then $n^6 + ... etc > 1$ so $(n+1) = p^2$ and $n^6 + ... etc = 1$ is out of the question.



              That just leaves $n + 1 = n^6 + .... = p$ as an option. And for that to work we must have $n = 1$. And in that case it does work out that $n+1 = n^6 + ... etc = 2$.






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$


                What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors.




                Or fewer.



                To rephrase and elaborate on greedoid's answer in a way that doesn't involve complete factoring:



                So if if the polynomial factor as $r(n)q(n)$ we must have either $r(n) = 1; q(n) = p^2$ for some $n$ and prime $p$ or $r(n) =q(n)=p$ for some $n$ and prime $p$.



                Note that if $n =-1$ we get the polynomial is $0$ so $n+1$ factors.



                And $frac {n^7 + n^6 + n^5 + 1}{n+1} = n^6 + n^4 -n^3 +n^2 -n + 1$.



                If $n+1 = 1$ then we get $n = 0$ and .... the other is $1$ and that won't do.



                It's pretty clear if $n > 1$ then $n^6 + ... etc > 1$ so $(n+1) = p^2$ and $n^6 + ... etc = 1$ is out of the question.



                That just leaves $n + 1 = n^6 + .... = p$ as an option. And for that to work we must have $n = 1$. And in that case it does work out that $n+1 = n^6 + ... etc = 2$.






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                $endgroup$




                What I found out is that this polynomial must be a square of prime number, otherwise it will have more than 3 divisors.




                Or fewer.



                To rephrase and elaborate on greedoid's answer in a way that doesn't involve complete factoring:



                So if if the polynomial factor as $r(n)q(n)$ we must have either $r(n) = 1; q(n) = p^2$ for some $n$ and prime $p$ or $r(n) =q(n)=p$ for some $n$ and prime $p$.



                Note that if $n =-1$ we get the polynomial is $0$ so $n+1$ factors.



                And $frac {n^7 + n^6 + n^5 + 1}{n+1} = n^6 + n^4 -n^3 +n^2 -n + 1$.



                If $n+1 = 1$ then we get $n = 0$ and .... the other is $1$ and that won't do.



                It's pretty clear if $n > 1$ then $n^6 + ... etc > 1$ so $(n+1) = p^2$ and $n^6 + ... etc = 1$ is out of the question.



                That just leaves $n + 1 = n^6 + .... = p$ as an option. And for that to work we must have $n = 1$. And in that case it does work out that $n+1 = n^6 + ... etc = 2$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 2 '18 at 21:23









                fleabloodfleablood

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