Totient number for which the smallest element of the inverse is even?

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Suppose , $k$ is a positive integer and there exists a positive integer $n$ with $varphi(n)=k$




  • Can the smallest value $n$ with $varphi(n)=k$ be even ?

  • Can all values $n$ with $varphi(n)=k$ be even ?



I did not find a counterexample even for the first statement upto $3cdot 10^4$. It is clear that the smallest value $n$ cannot be of the form $4m+2$ because then $fracn2$ would also be a possible value.



The given statement is a strengthening of the conjecture that there is no $k$ such there is exactly one $n$ with $varphi(n)=k$.




number-theory elementary-number-theory totient-function




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    up vote
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    Suppose , $k$ is a positive integer and there exists a positive integer $n$ with $varphi(n)=k$




    • Can the smallest value $n$ with $varphi(n)=k$ be even ?

    • Can all values $n$ with $varphi(n)=k$ be even ?



    I did not find a counterexample even for the first statement upto $3cdot 10^4$. It is clear that the smallest value $n$ cannot be of the form $4m+2$ because then $fracn2$ would also be a possible value.



    The given statement is a strengthening of the conjecture that there is no $k$ such there is exactly one $n$ with $varphi(n)=k$.




    number-theory elementary-number-theory totient-function




    share|cite|improve this question





















      up vote
      3
      down vote

      favorite
      3









      up vote
      3
      down vote

      favorite
      3






      3





      Suppose , $k$ is a positive integer and there exists a positive integer $n$ with $varphi(n)=k$




      • Can the smallest value $n$ with $varphi(n)=k$ be even ?

      • Can all values $n$ with $varphi(n)=k$ be even ?



      I did not find a counterexample even for the first statement upto $3cdot 10^4$. It is clear that the smallest value $n$ cannot be of the form $4m+2$ because then $fracn2$ would also be a possible value.



      The given statement is a strengthening of the conjecture that there is no $k$ such there is exactly one $n$ with $varphi(n)=k$.




      number-theory elementary-number-theory totient-function




      share|cite|improve this question











      Suppose , $k$ is a positive integer and there exists a positive integer $n$ with $varphi(n)=k$




      • Can the smallest value $n$ with $varphi(n)=k$ be even ?

      • Can all values $n$ with $varphi(n)=k$ be even ?



      I did not find a counterexample even for the first statement upto $3cdot 10^4$. It is clear that the smallest value $n$ cannot be of the form $4m+2$ because then $fracn2$ would also be a possible value.



      The given statement is a strengthening of the conjecture that there is no $k$ such there is exactly one $n$ with $varphi(n)=k$.





      number-theory elementary-number-theory totient-function





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      share|cite|improve this question




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      asked Jul 30 at 7:30









      Peter

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          See OEIS sequence A002181. The answers to both questions are yes. For example $varphi(n)=16842752$ only for even $n$.






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            1 Answer
            1






            active

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            active

            oldest

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            active

            oldest

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            up vote
            4
            down vote













            See OEIS sequence A002181. The answers to both questions are yes. For example $varphi(n)=16842752$ only for even $n$.






            share|cite|improve this answer

























              up vote
              4
              down vote













              See OEIS sequence A002181. The answers to both questions are yes. For example $varphi(n)=16842752$ only for even $n$.






              share|cite|improve this answer























                up vote
                4
                down vote










                up vote
                4
                down vote









                See OEIS sequence A002181. The answers to both questions are yes. For example $varphi(n)=16842752$ only for even $n$.






                share|cite|improve this answer













                See OEIS sequence A002181. The answers to both questions are yes. For example $varphi(n)=16842752$ only for even $n$.







                share|cite|improve this answer













                share|cite|improve this answer



                share|cite|improve this answer











                answered Jul 30 at 8:01









                Robert Israel

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