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

zbi74583.boat at orange.zero.jp zbi74583.boat at orange.zero.jp
Wed Jan 14 07:38:27 CET 2009

```    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   　....

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

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