smarandache function (book series), vol. 6

8/9/2019 SMARANDACHE FUNCTION (book series), Vol. 6

http://slidepdf.com/reader/full/smarandache-function-book-series-vol-6 1/74

SM R



Authors o f paper



O

Marcela P



= [ S ( ~ ) -

S

B l l ~

S(pf)

~ S pf-l)

for

e



In

)]

it is proved

the

fo

5 l), wher

p is a prime

na.

r sing this formula., we heLve



PROPERTIE

b

Depart



Proof First

we

cons

x = 2

we

obtain Fs 2) = 5

Next we shall write th



F(

) S

x+y).

X

+

Y

=

PI

T 'T

But

from

(1) we

have

t



We consider now a

> 1

(n)

L

We try to find lim - - I = : :

(n)

Let

an = L P, - ;r(n)



ON A L

N

by



But ~ 2



On

the oth

consequently:

r. S

(k)

~

( k+ l )



R ~ F E R E N E S

1. Smarandache Func

Vol .4-5

1994,

Nu



S

O

M

E

?

R

O

P

E

R

T

I

a

a



It r e su l t . s t hat S k

s ing . Let

l I

so : )

n

1



where 0 S t S p -

The p ~ o e d u ~ e o f



q

~ ~ r e s u l ~

~ h ~

E

lc=1



~ S p i

)

~ i

~ S p

Because have

S \.Cn



SOME

P



e

have

S(P

ra t io of S(p;)/(P p



a,

a,

d

contains in f in i te ly

Proof

of claim:



and

using

d • 100

10D



Proof

I t is well-kno



(2 2

. . . : ; . . .

2 2 2

And applying

~ l l

yie ld a valu



p - p

Now,

examine



S p) ,S p ·p )

,S

will

contain

a l l

posi t i

Proof

I t has already



so we proceed to ca

However,

i t i s

numbers

that

i f

m

then mop

i s

the

sma



Corollary:

Let

n be a posi

nand

k

another posi



5

E

.:. 2

k·+

1

where s

is

the

num

integer such that



such

tha t

mp

= n

gr

where

k i s the

expon

grows

on

the order o

of items in the

prod



n=265225=5 'S ' :03 '10

n=84363

n=84363



8 4900)

8 4901)

=

8

8 3211b)

8 32111)

=

8 368138) 8 368139)

8 415664)

i



~ h t the resolut ion

perhaps

in f in i te ly



A

B

O

U

C

I

on

B

I

Depa

1



/J .

x) = p .1:. 1

2



I

if

a/ is odd



1)

Find the solutions o

If

m is

not

a

perfect sq

If m

is

a

perfect squ

equation

xa

(x)

=

Z2.



u>v, u, v)=1 u and v h

xa x) =

12(U:

+V

ya y) = I: U

Z

+

za z) =

12(U

2

- V

indicated

in

the first pro



a x)

=

3s+n

a y)

=

-m - 2s

a z)=2n+3m, m n

For

m

= 1,

n

=2,

S

=

(2a

2

y2), a ~ y EZ·.

solutions.



S ME REM

. /OF

TO



For k

= .. ,

q

2,

the

p

between

S(p

.... 1 and S(p).

q q l

H o w e v ~ r , if

there i a numb

has solutions. This condit



r

> can

take o

,S r) J

o<

: _ r_ :

< ; S r)

l (

S r)

, r

I

Sex) _ _

=

S pa)2 >X ~



:

We have O < ~ 5-

. x, r«

. 3 .

+ =>

52 ,xl

<

9, but this is

5 ,xl

Ifp=7

=>

S(x)=S(73)=

(.. X

I.

h

0

We ave

< < ~ <



Let x be a solution

We have S(x)

is

divisible by m, so (S(x

(rnx). We know that S(



If the equation has a solu

If m=n, the equation bec

and any prime number as we

solutions are x= 1 and x=2.



SO

INSP



def ined

by

mVn=m

0 ({.,}) =l=ey , then

1.PROPOSITION. Let



e

have

VS- a)

=aVb=h



CAl

Nt / P

s

) 2 - - ~ N

C l

o

: N-/p

s

) 2 - - ~ N

Moreover, N*/P



and

C }()(flJ}) €G i f

Thatis

1 ,

In fac t the



SMARANDAC

by I

Depar



O

b

vi

o

u

sl

y,

.

2

)

SI



5. Theorem [s

(N ,.)

in ( N , ~ , + )

and

(N °,·)

in N°



Let p - l =

qj > p for j

el t.

It

follows

S(p



T

HE PRO

Univ



Ifp is

a

prime num

equation =

If q

s

a composite



For x = y we ha.ve an equ

the

Lipschitz condition

Rema.rk

3.

In

it

is

.



Proposition 4

The

~ " s l ' ~ ; t z

1 ' 1 ) , " " ; f ' , ~ "

-I ~ . . .

-'

-

- •.

P ~ ~ , ; f .

(Compa.re



We

ha.ve:

Tndeed, if

.:t

= 1, then



IUD'

BIS'rO

ADDENDA

(III) :



cover

IV,

[106] Pe te r B

Losungen ,

Vol.

49, N

[107] Gh. Tomo

impotr iva

Ed.

Macar



1993;

reviewed in <

11-06:

[131]

Gh.

Stroe

Bucharest ,

A

[132]

Dr. Dumi

Smarandache



M a t e m a t i c ~ >

p . 43;

[148]

A.

Stuparu

Matematica

p .

43;

[149] Pedro

Me

Chisinau, R



PROP



A GENERAL

Let n be a

composi te



N

IMPORT NT

FORM



Henry Ibstedt, Smarandach

Marcela Popescu, Paul Popes

col lec t ion

of pa



numbers

sequences i

qeometries alqebraic

and

loq ic etc .

Dr.

C Dum it r e s cu

Depar tment o f

Ma

Un ive r s i t y

o f

Cra

smarandache function (book series), vol. 6

Documents