[seqfan] UnitaryPhi(m)=UnitaryPhi(n)=(3*m^(1/2)-2*n^(1/2))^2

[zbi74583.boat at orange.zero.jp]

    Hi, Seqfans

    I considered the numbers m,n such that UnitaryPhi(m)=
UnitaryPhi(n)=(3*m^(1/2)-2*n^(1/2))^2 or UnitaryPhi(m)=
UnitaryPhi(n)=k*(3*m^(1/2)-2*n^(1/2))^2 1<k .

    At first I had thought that no example exists.
    Because see the following equation.

         UnitaryPhi(m)= UnitaryPhi(n)=1/4*(m^(1/2)+ n^(1/2))^2   　....[1]

    Only one example m=n=1 exists
    If m=n then
         UnitaryPhi(m)=m
         For 1<m   UnitaryPhi(m)<m
         So no other example exists
    And
    If m,n are different then
          UnitaryPhi(m)  ^(1/2) = 1/2*(m^(1/2)+ n^(1/2)) < m^(1/2)
         n^(1/2) < m^(1/2)
    Samely
         m^(1/2) < n^(1/2)
    It is impossible.

    The case of UnitaryPhi(m)= UnitaryPhi(n)=(3*m^(1/2)-2*n^(1/2))^2 or
UnitaryPhi(m)= UnitaryPhi(n)=k*(3*m^(1/2)-2*n^(1/2))^2 .

    If m=n then

         UnitaryPhi(m)=m
         UnitaryPhi(m)=k*m  1<k

    It seems to be the same as the equation 1.
         m=1 and no example for second

    Bur if m,n are different then

         UnitaryPhi(m) ^(1/2)= 3*m^(1/2)-2*n^(1/2)<m^(1/2)
                    m<n
         UnitaryPhi(n) ^(1/2)= 3*m^(1/2)-2*n^(1/2)<n^(1/2)
                    m<n

    It is possible.
    In fact several examples exist.

    I got one example for k=2/7 of the following equation.

         UnitaryPhi(m)= UnitaryPhi(n)=k*(5*m^(1/2)-3*n^(1/2))^2 .
         m=2^6*3^2*5^2*7^4
         n=2^6*3^4*7^2*11^2

    If m=n then

         UnitaryPhi(m)=8/7*m

    It is also impossible.
    I wonder what is the maximal of k whose equation has one example.



    %I A000001
    %S A000001 10147737600, 166240237363200, 310334052188160, 9500311941120000
    %N A000001 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(n)=
(3*m^(1/2)-2*n^(1/2))^2, m<n
               Sequence gives n.
    %C A000001 If m=n then only one example m=n=1 exists.
    %e A000001 Factorization :
               2^12*5^2*7*13*3^2*11^2
               2^13*5^2*7*13*8191*3^2*11^2
               2^14*5*7^2*13*43*127*3^2*11^2
               2^15*5^4*7*13*31*151*3^2*11^2
    %Y A000001 A000002
    %K A000001 none
    %O A000001 0,1
    %A A000001 Yasutoshi Kohmoto zbi74583.boat at orange.zero.jp

    %I A000002
    %S A000002 8954982400, 146700521676800, 273857689763840, 8383654522880000
    %N A000002 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(n)=
(3*m^(1/2)-2*n^(1/2))^2, m<n
               Sequence gives m.
    %C A000002 If m=n then only one example m=n=1 exists.
    %e A000002 Factorization :
               2^12*5^2*7*13*31^2
               2^13*5^2*7*13*8191*31^2
               2^14*5*7^2*13*43*127*31^2
               2^15*5^4*7*13*31*151*31^2
    %Y A000002 A000001
    %K A000002 none
    %O A000002 0,1
    %A A000002 Yasutoshi Kohmoto zbi74583.boat at orange.zero.jp



    Yasutoshi