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PureMathematics
n
Ø
ê
Æ
,2019,9(9),1102-1107
PublishedOnlineNovemb er2019inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2019.99135
SolutionsofaClassofEulerFunction
Equations
KeliPu
DepartmentofMathematicsandCom puterScience,ABaTeachersCollege,WenchuanSichuan
Received:Nov.2
nd
,2019;accepted:Nov.21
st
,2019;published:Nov.28
th
,2019
Abstract
Let
n
beapositiveinteger,
ϕ
(
n
)
isEulerfunction,thevalueisequaltothesequence
0
,
1
,
2
,...,n
−
1
whichareprimeto
n
.Infact,discussingthesolutionsofEulerfunction
equationisameaningfulwork,moreover,thepropertiesofthefunctionarevery
importanttodiscussthesolution.Inthispaper,usingthepropertiesoftheEuler
function,wediscussthenecessityofintegersolutionoftheEulerfunctionequation
ϕ
(
mn
)=
aϕ
(
m
)+
bϕ
(
n
)+
c
,andthengivesallsolutionsifa=5,b=6,C=16.
Keywords
EulerFunction,PropertiesofEulerFunction,IntegerSolution
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c
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c
11
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µ
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c
11
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©
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Ú
^
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Æ
Œ
s
.
˜
a
î
.
¼
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)
[J].
n
Ø
ê
Æ
,2019,9(9):1102-1107.
DOI:10.12677/pm.2019.99135
Æ
Œ
s
Á
‡
n
´
ê
,
ϕ
(
n
)
´
Í
¶
î
.
¼
ê
§
§
Š
u
S
0
,
1
,
2
...n
−
1
¥
†
n
p
ƒ
ê
‡
ê
"
é
u
9
î
.
¼
ê
•
§
)
?
Ø
´
˜
‡L
k
¿Â
‘
K
§
î
.
¼
ê
5
Ÿ
3
?
Ø
î
.
¼
ê
•
§
)
¥–
'
-
‡
"
©
|
^
î
.
¼
ê
5
Ÿ
ƒ
'
(
Ø
§
?
Ø
˜
a
î
.
¼
ê
•
§
ϕ
(
mn
)=
aϕ
(
m
)+
bϕ
(
n
)+
c
•
3
ê
)
7
‡
^
‡
§
¿
‰
Ñ
a
=5
,b
=6
,c
=16
ž
§
T
î
.
¼
ê
•
§
Ü
)
"
'
…
c
î
.
¼
ê
,
î
.
¼
ê
5
Ÿ
,
ê
)
Copyright
c
2019byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.
Ú
ó
n
´
ê
§
ϕ
(
n
)
´
Í
¶
î
.
¼
ê
§
§
Š
u
S
0
,
1
,
2
...n
−
1
¥
†
n
p
ƒ
ê
‡
ê
"
'
u
Euler
¼
ê
ϕ
(
n
)
•
§
´ê
Ø
¥
š
~
-
‡
Ú
k
¿Â
‘
K
§
é
k
'
u
ϕ
(
n
)
5
Ÿ
Ú
ϕ
(
n
)
k
'
Ø
½
•
§
ï
Ä
§
N
õ
Æ
ö
?
1
&?
(
Œ
ë
w
©
z
[1–13])
"
Ù
¥
©
z
[1,2]
é
u
•
§
ϕ
(
x
)=
m
)?
1
?
Ø
§
©
z
[3]
‰
Ñ
m
=2
p,
2
p
n
,
2
pq
ž
•
§
ϕ
(
x
)=
m
)
(
Ù
¥
p,q
•
ƒ
ê
§
n
•
ê
)
"
3
©
z
[4–7]
¥
K
©
O
?
Ø
•
§
ϕ
(
mn
)=
k
(
ϕ
(
m
)+
ϕ
(
n
))
Ù
¥
k
∈
Z
§
3
k
Ø
Ó
Š
ž
)
"
é
/
X
ϕ
(
mn
)=
aϕ
(
m
)+
bϕ
(
n
)+
c
î
.
¼
ê
š
‚
5
•
§
§
©
z
[13]
‰
Ñ
a
=7
,b
=8
,c
=16
ž
•
§
Ü
)
"
©
?
Ø
•
§
ϕ
(
mn
)=
aϕ
(
m
)+
bϕ
(
n
)+
c
(
Ù
¥
a,b,c
∈
Z
)
ê
)
§
¿
‰
Ñ
ϕ
(
mn
)=5
ϕ
(
m
)+6
ϕ
(
n
)+16
Ü
ê
)
"
2.
î
.
¼
ê
5
Ÿ
ƒ
'
(
J
Ú
n
1
[14]
m,n
•
?
¿
ê
§
e
m
|
n
§
K
ϕ
(
m
)
|
ϕ
(
n
)
"
Ú
n
2
[14]
é
?
¿
ê
m,n
§
e
gcd
(
m,n
)=
d
§
K
ϕ
(
mn
)=
dϕ
(
m
)
ϕ
(
n
)
ϕ
(
d
)
"
Ú
n
3
[14]
n
≥
1
ž
§
ϕ
(
n
)
≤
n
§
n
≥
3
ž
§
ϕ
(
n
)
7
•
ó
ê
"
í
Ø
4
é
?
¿
n
‡
ê
x
1
,x
2
,...,x
n
§
k
ϕ
(
x
1
x
2
...x
n
)
≥
ϕ
(
x
1
)
ϕ
(
x
2
)
...ϕ
(
x
n
)
DOI:10.12677/pm.2019.991351103
n
Ø
ê
Æ
Æ
Œ
s
y
²
d
Ú
n
2
9
Ú
n
3
á
"
Ú
n
5
[15]
é
?
¿
ê
n,p
´
ƒ
ê
§
K
ϕ
(
np
)=
(
(
p
−
1)
ϕ
(
n
)
,
(
n,p
)=1
,
pϕ
(
n
)
,
(
n,p
)=
p.
3.
Ì
‡
(
J
9
y
²
Ú
n
6
[3]
p
•
ƒ
ê
§
ϕ
(
x
)=2
p
)
•
(1)
p
=2
ž
§
x
=5
,
8
,
10
,
12
(2)
p
=3
ž
§
x
=7
,
9
,
14
,
18
(3)
p
≥
5
ž
§
g
=2
p
+1
•
ƒ
ê
§
ϕ
(
x
)=2
p
k
ü
‡
)
x
=
g,
2
g
¶
g
=
ϕ
(2
p
+1)
Ø
•
ƒ
êž
§
K
ϕ
(
x
)=2
p
Ã
ê
)
"
Ú
n
7
[3]
e
ϕ
(
x
)=2
,
K
x
=3
,
4
,
6
"
ϕ
(
x
)=2
2
§
K
x
=5
,
8
,
10
,
12
.
ϕ
(
x
)=2
3
§
K
x
=15
,
16
,
20
,
24
,
30
.
ϕ
(
x
)=2
4
§
K
x
=17
,
32
,
34
,
40
,
48
,
60
.
ϕ
(
x
)=2
5
§
K
x
=51
,
64
,
68
,
80
,
96
,
102
,
120
.
½
n
6
î
.
¼
ê
•
§
ϕ
(
mn
)=
aϕ
(
m
)+
bϕ
(
n
)+
c
(
Ù
¥
a,b,c
∈
Z
)
§
e
•
3
ê
)
(
m,n
)
§
K
ϕ
(
gcd
(
m,n
))
|
c
"
y
²
Ø
”
gcd
(
m,n
)=
d
§
K
d
|
m,d
|
n
"
d
Ú
n
1
Œ
ϕ
(
d
)
|
ϕ
(
m
)
…
ϕ
(
d
)
|
ϕ
(
n
)
"
-
ϕ
(
m
)=
m
1
ϕ
(
d
)
,
ϕ
(
n
)=
n
1
ϕ
(
d
)
,
Ù
¥
m
1
,n
1
∈
Z
+
§
2
d
Ú
n
2
ϕ
(
d
)(
dm
1
n
1
−
am
1
−
bn
1
)=
c
§
=
ϕ
(
d
)
|
c
§
y
"
½
n
7
î
.
¼
ê
•
§
ϕ
(
mn
)=5
ϕ
(
m
)+6
ϕ
(
n
)+16
ê
)
21
|
§
©
O
•
(
m,n
)=(53
,
7)
,
(53
,
9)
,
(53
,
14)
,
(53
,
18)
,
(106
,
7)
,
(106
,
9)
,
(15
,
29)
,
(16
,
29)
,
(20
,
29)
,
(24
,
29)
,
(30
,
29)
,
(15
,
58)
,
(8
,
38)
,
(8
,
54)
,
(10
,
38)
,
(10
,
54)
,
(12
,
38)
,
(75
,
12)
,
(12
,
18)
,
(20
,
10)
,
(30
,
10)
.
y
²
-
gcd
(
m,n
)=
d
§
d
Ú
n
2
Œ
ϕ
(
d
)(
dm
1
n
1
−
5
m
1
−
6
n
1
)=16
§
2
d
½
n
6
Œ
ϕ
(
d
)
|
16
§
K
ϕ
(
d
)=1
,
2
,
4
,
8
,
16
"
e
¡
©
œ
¹
?
Ø
µ
I.
ϕ
(
d
)=1
§
K
d
=1
½
2
"
d
=1
ž
§
K
m
1
n
1
−
5
m
1
−
6
n
1
=16
§
=
(
m
1
−
6)(
n
1
−
5)=46
d
d
Œ
(
m
1
,n
1
)=
(7
,
51)
,
(52
,
6)
,
(8
,
28)
,
(29
,
7)
§
(
m
1
,n
1
)=(7
,
51)
,
(29
,
7)
ž
§
du
ϕ
(
d
)=1
§
¤
±
ϕ
(
m
)
ϕ
(
n
)
þ
•
Û
ê
§
ù
†
Ú
n
3
g
ñ
§
¤
±
(
m
1
,n
1
)=(52
,
6)
,
(8
,
28)
"
(
m
1
,n
1
)=(52
,
6)
ž
§
d
ž
ϕ
(
m
)=52
§
DOI:10.12677/pm.2019.991351104
n
Ø
ê
Æ
Æ
Œ
s
K
O
Ž
Œ
m
=53
,
106
§
ϕ
(
n
)=6
§
K
n
=7
,
9
,
14
,
18
,
¤
±
(
m,n
)=(53
,
7)
,
(53
,
9)(53
,
14)
,
(53
,
18)
,
(106
,
7)
,
(106
,
9)
"
(
m
1
,n
1
)=(8
,
28)
ž
§
ϕ
(
m
)=8
,
K
m
=15
,
16
,
20
,
24
,
30
"
ϕ
(
n
)=28
§
K
n
=
29
,
58
"
¤
±
(
m,n
)=(15
,
29)
,
(16
,
29)(20
,
29)
,
(24
,
29)
,
(30
,
29)
,
(15
,
58)
.
d
=2
ž
§
K
2
m
1
n
1
−
5
m
1
−
6
n
1
=16
§
=
(
m
1
−
3)(2
n
1
−
5)=31
,
d
d
Œ
(
m
1
,n
1
)=
(4
,
18)
,
(34
,
3)
"
(34,3)
Ø
÷
v
^
‡
§
"
=
(
m
1
,n
1
)=(4
,
18)
§
K
ϕ
(
m
)=4
§
K
m
=5
,
8
,
10
,
12
"
ϕ
(
n
)=18
§
K
n
=19
,
27
,
38
,
54
"
¤
±
(
m,n
)=(8
,
38)
,
(8
,
54)
,
(10
,
38)
,
(10
,
54)
,
(12
,
38)
.
II.
ϕ
(
d
)=2
§
K
d
=3
,
4
,
6
"
d
=3
ž
§
K
3
m
1
n
1
−
5
m
1
−
6
n
1
=8
§
=
(3
m
1
−
6)(3
n
1
−
5)=54
"
K
(
m
1
,n
1
)=(20
,
2)
§
d
ž
ϕ
(
m
)=40
§
K
m
=41
,
55
,
75
,
82
,
88
,
100
,
110
,
132
,
150
,ϕ
(
n
)=4
§
K
n
=5
,
8
,
10
,
12
"
¤
±
(
m,n
)=(75
,
12)
d
=4
ž
§
K
4
m
1
n
1
−
5
m
1
−
6
n
1
=8
§
=
(2
m
1
−
3)(4
n
1
−
5)=31
§
K
(
m
1
,n
1
)=(2
,
9)
§
d
ž
ϕ
(
m
)=4
§
K
m
=5
,
8
,
10
,
12”
ϕ
(
n
)=18
,
K
n
=19
,
27
,
38
,
54
"
Ï
gcd
(
m,n
)=4
§
¤
±
•
§
Ã
ê
)
"
d
=6
ž
§
K
6
m
1
n
1
−
5
m
1
−
6
n
1
=8
§
=
(2
m
1
−
2)(6
n
1
−
5)=26
§
K
(
m
1
,n
1
)=(14
,
1)
,
(2
,
3)
"
(
m
1
,n
1
)=(14
,
1)
§
d
ž
ϕ
(
m
)=28
§
K
m
=29
,
58
.ϕ
(
n
)=2
§
K
n
=3
,
4
,
6
§
gcd
(
m,n
)=6
§
d
ž
•
§
Ã
ê
)
"
(
m
1
,n
1
)=(2
,
3)
§
d
ž
ϕ
(
m
)=4
§
K
m
=5
,
8
,
10
,
12
.ϕ
(
n
)=6
§
K
n
=
7
,
9
,
14
,
18
"
¤
±
(
m,n
)=(12
,
18)
.
III.
ϕ
(
d
)=4
§
K
d
=5
,
8
,
10
,
12
"
d
=5
ž
§
5
m
1
n
1
−
5
m
1
−
6
n
1
=4
§
=
(5
m
1
−
6)(
n
1
−
1)=10
§
d
ž
Ø
•
3
m
1
,n
1
∈
Z
+
¦
ª
¤
á
§
•
§
Ã
ê
)
"
d
=8
ž
§
8
m
1
n
1
−
5
m
1
−
6
n
1
=4
§
=
(4
m
1
−
3)(8
n
1
−
5)=31
§
d
ž
Ø
•
3
m
1
,n
1
∈
Z
+
¦
ª
¤
á
§
•
§
Ã
ê
)
"
d
=10
ž
§
10
m
1
n
1
−
5
m
1
−
6
n
1
=4
§
=
(5
m
1
−
3)(2
n
1
−
1)=7
§
K
(
m
1
,n
1
)=(2
,
1)
§
d
ž
ϕ
(
m
)=8
§
K
m
=15
,
16
,
20
,
24
,
30
.ϕ
(
n
)=4
§
K
n
=5
,
8
,
10
,
12
"
¤
±
(
m,n
)=(20
,
10)
,
(30
,
10)
.
d
=12
ž
§
12
m
1
n
1
−
5
m
1
−
6
n
1
=4
§
=
(2
m
1
−
1)(12
n
1
−
5)=13
§
d
ž
Ø
•
3
m
1
,n
1
∈
Z
+
¦
ª
¤
á
§
•
§
Ã
ê
)
"
DOI:10.12677/pm.2019.991351105
n
Ø
ê
Æ
Æ
Œ
s
IV.
ϕ
(
d
)=8
§
K
d
=15
,
16
,
20
,
24
,
30
"
d
=15
ž
§
15
m
1
n
1
−
5
m
1
−
6
n
1
=2
§
=
(5
m
1
−
2)(3
n
1
−
1)=4
.
d
=16
ž
§
16
m
1
n
1
−
5
m
1
−
6
n
1
=2
§
=
(8
m
1
−
3)(16
n
1
−
5)=31
.
d
=20
ž
§
20
m
1
n
1
−
5
m
1
−
6
n
1
=2
§
=
(10
m
1
−
3)(4
n
1
−
1)=7
.
d
=24
ž
§
24
m
1
n
1
−
5
m
1
−
6
n
1
=2
§
=
(24
m
1
−
5)(4
n
1
−
1)=13
.
d
=30
ž
§
30
m
1
n
1
−
5
m
1
−
6
n
1
=2
§
=
(5
m
1
−
1)(6
n
1
−
1)=3
.
d
ž
§
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ϕ
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d
)=16
§
K
d
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,
32
,
34
,
40
,
48
,
60
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Ó
þ
O
Ž
Œ
§
þ
Ø
•
3
m
1
,n
1
¦
ª
dm
1
n
1
−
5
m
1
−
6
n
1
=1
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á
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•
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z
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ϕ
(
x
)=
m
.
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,
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https://doi.org/10.2307/121103
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‰
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r
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o
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†
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ü
‡
•
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Æ
¢
‚
†
@
£
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[7]
°
•
,
p
w
,
…
Ê
.
k
'
Euler
¼
ê
ϕ
(
n
)
•
§
Œ
)
5
¯
K
[J].
ô
Ü
‰
Æ
,2016,34(1):
15-16.
[8]
•
R
.
˜
a
•
¹
Smarandache
¼
ê
Ú
Euler
¼
ê
•
§
[J].
Ü
H
Œ
ÆÆ
(
g
,
‰
Æ
‡
),2012,
34(2):70-73.
[9]
4
R
,
œ
+
.
'
u
•
§
P
S
(
d
)=
ϕ
(
n
)
Œ
)
5
.
Ü
H
Œ
ÆÆ
(
g
,
‰
Æ
‡
),2013,35(6):54-58.
[10]
•
I
¦
.
˜
‡
•
¹
Euler
¼
ê
•
§
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ê
Æ
†A^
ê
Æ
,2007,23(4):439-445.
[11]
Ü
o
,
Ú
k
•
.
˜
‡
•
¹
Euler
¼
ê
•
§
ê
)
[J].
u
¥
“
‰
Œ
ÆÆ
(
g
,
‰
Æ
‡
),2015,
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ϕ
(
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).
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n
Ø
ê
Æ
Æ
Œ
s
[13]
g
Ÿ
#
#
Ü
,
Ü
o
,
=
÷
Œ
.
˜
‡
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'
Euler
¼
ê
ϕ
(
n
)
š
‚
5
•
§
)
[J].
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Œ
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[15]
š
}
•
,
§
œ
.
'
u
•
§
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(
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(
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n
Ø
ê
Æ