.

Probleuesessiou (Chapter G) section G - I

Towler's ol - functionDef locus is the size of reduced residue system modulo n

4G) = I

10 is a multiplicative function : if g.e. d. Cue , u) =L , then oleum) -- focusedcue)ofCpu) -- pm- pm

-'

= put- '

Cp - i) -- pull - Ip)

941=41,14 - f) (th G - 2)

① If olcmskm - i) then mis square-free

If Assume otherwise , palm .

Then we -_ pth with t>2 and plan ( g.c. d. Cpt, n) =L)&CmI= ofCpt ) par) = pt

-

Kp-Dolen).

Since tza,

t - I > I,therefore plotcue)

At the same time weocueodp) implies we-E - I Cueodp) , thus pXCupl gem)

plan. .)} imply Kusum..)

⑨ If in >2,

then dem) is even . I Ole't =L,& Ca) - I , 4933=2 . . .

PI Either there is an odd prime plan orheat

,

t > 2.

a) Let plue , p is an odd prime . Then we=psu, g. e. d. Cps. a) = I . s > I

loan) - ofCps> peu) =p'"

den).Since p is odd

, p - l is even,LICp- i)

Thus 2114cm), gene) iseveu

b) Let in -- It, t >2 .

4cm) = It- '

(2 - c) = It- '

is because t- I > I.

461=1 has solutions n -- I,h=2

,

and no other solutions / The function of is"sestet

"

increasing

01423=4 41,5, 7,11 }Prep For positive integers ok

,l da) -- 6

peke) > 4lb) If us.in,

then

P1 h- pi'

. . - pit ke -- pi: .. pig ,

"

. . - g! u. > ri945) > again )

lock) -- I? pep locket -- 1744 xolegii . - qui )since Vissi . 44!) =p?

-'

Cp.

- is > pCp÷) -- of Cp! )- -

④ Ioann) = focus Answer : fuel, any ng

① Assume g.e. d. Cue , uh ,hw=2,hisodd€

&em¥=o#01cm) =L implies cuter -2

② Assure g.e. d. Cue > u) > I .

There is a primepsuehthatpln.pvu.h-pk.pxkocut-ocp.jo/Ck) Okun) - den)

we -- psl. take s - o / TIECK) =p's-'Cpque)

can -- petskb , pxkb often) - depots > duel ) lock) . psyche )This cannot happen because

psst and lock) ⇐ locker )

⑥ 447-12 / 1045=1,7 (pm- put'

) - 17pm-'

(p - i)In plan

Preparations : simpler equations up! .- pins447=1 solutions : h - I

,n=2-

n--2^36 ten)=2" 3£! 2 a> o

,

boopen)=2 solutions : u =3

,h=4

,u=6

a-_ I 6=1 h=6 46) =L0/41=3 no solutions a-- o 8=1 n =3 lots) -- 2

a=2 6=0 h=4 0163=2

-

h=p?. . p.es If pi > 13 , then p - Isla

The only possible psiuee factors of u are :

2,3, 5,7 , 11,13

Iet 13 be in, namely a- Been , 13km

tech> = 13--143- c) dew) = 12

e - I, 4cm) =/ we -_ I h=l3①

bred h-②Now 13 is out

.

Let II be in, namely n - hem 11km

pen) -- Ile- '

(Il - 1) dem) = 12

this cannot happen because 11-1=10 is divisible by 5while 12 is not divisible by 5How It is out

Let 7 be in, namely h = Few 7km

Ofcu) = 7-e- '

(7- c) of Cue) = 12

E - I.

6 of cue) = 12

& fu) =L we =3,Ll,6

Solutions

h=7 h=28 h=76-Now It is out

Let 5 be in, namely h=5em

,3- Xue

peu) . b-e- '

(5 - 1) offer) = 12

e -- I 4/0 Cue) = 12

dem)=3There is no such eee

.

How 5 is out

Let now h - 2-38 wanted : a and to such that 46-39--12.

Assume a=o. Of (3h ) = 3k! (3-1)=12 3h- '

a 6 -never happens

Assume 8=0. & a) = La

- '

(2-1)=12 La- '

= 12 - never happens

Assume a> I,b > I

ofcgazeo ) = La- '

(2- e) 3h- '

(3-1)=2936- '

= 12=273"

4%2h=2d3d=3①

Answer : Oleh) = 12 for h= 13, 26,21 , 28,42

,

36-

8,9100 Alternative proof of da) = uphill - f) (Theo-a)⑨ The formula comes from :

• olep"

> =p"

- p"'

• 01 is multiplicative ( follows from THE-7 i ol is multiplicative asa member of Mobius pair Cn, lo) n = Ign&Cd)

byTh G - c )⑧Let g.c. d. Cue , u) - I

5=4 nxtuey I { all values in reduced residue system modulo nm )S is reduced residue system modulo wenn#

⑨ #5=4Cue) docu) # 5=01 Cuen)

Pf of ⑧-

To provev l

.For a ES

, g.c. d. (a, we a) = I

✓ 2.If a is an integer such that g.c.d. Lateen) =L , thenthere exist x and y

suck that a = *tiny duodena)v 3

.

all elements of 5 are paireois distinct modulo we u

① To pseueigedcnxtmy.mu)- IIet plum Wanted : pxcuxtuey)Assume plea Pth. Then hxtuey= hx # ocueodp) pkx because

× is an element ofa Seed residue systemmodulo in

, plus

② Wanted : to solve the congruenceso g.e.d. Cx, w) - I

a=hX they(weed can) for x and y

foray a such that g.c. d. (a. urn) - I

Modulo lie : a = nx (mod w) - has a uniqueueodulo ur solution

by Th 5- I because g.c.d. Cn , we>=Lluodulo u : a = my Ed")

eestaiuly, g.c. d. Gin) - lbecause

g.c. d. (que) =L.

Leta - nx -my ⇒ reared un) Ts 8=0 Cured mu) ? - Yes

.

lllodulo eee : V = a - ux = • (mod .lu)lkodulo n : r= a - my= @ Cueodu)) imply 8=0 Canadian)

because g.e.d. (en , a) El

③ Assume wanted :

""otago Mkt my, Emad can)

Xo -=X , (modem) I

gooey,Cueodu)hex. -x ,)=w5y,-g.) duodena)

In particular ,hcx.

- x,)=o (modern)

bet hcx. - x .),g.c. d. Cue >

u)=l

Th 'd-3 implies eukx. - x .)Xo ⇒ X. (ueodue)