Smarandache Function Journal, 1

Vol. 1, 1990

ISSN 1053-4792

SMARANDACHE FUNCTION JOURNAL

S(n) is the smallest integer such that S(n)! is divisible by n S(n) is the smallest integer such that S(n)! is divisible by n S(n) is the smallest integer such that S(n)! is divisible by n

Number Theory Company

1

Editorial

Flor entin Smar andac he, a lliath emat ician from East ern Euro pe, escap ed fran his coun try beca use the comm unist auth oriti es had proh ibite d the publ icati on of his

resea~ch

pape rs and his part icipa tion in inter natio nal cong resse s.

Afte r two year s of wait ing in a poli tica l

~efugee

camp in

Turk ey, he emig rated to the unite d state s.

As resea rch work ers, rece iving our co-w orke r, we decid ed to publ ish a selec tion of his pape rs.

R. Mull er, Edit or

Read ers are enco urage d to subm it to the

Editor

manu scrip ts conc ernin g this func tion and/ or its prop ertie s, relat ions , appl icati ons, etc.

A profo und know ledge of this func tion woul d cont ribu te to the study of prim e numb ers, in acco rdan ce with the

fo~lowing

prop erty:

It p is a numb er grea ter

than 4, then p is prim e it and only if ry (p)

= p.

The manu scrip ts may be in the form at of rema rks, conj ectu res,

(un)s olved and/ or open prob lems , note s,

resea rch pape rs, etc.

INDICATION TO AUTHORS Authors of papers concerning any of Smarandache type functions are encouraged to submit manuscripts to the Editors: Number Theory Company c/o R. Muller 2220 W. Bloomfield Ave. Phoenix, AZ 85029, USA The submitted manuscripts may be in the format of remarks, conjectures, solved/unsolved or open new proposed problems, notes, articles, miscellaneous, etc.

They must be original work and camera ready [typewritten/computerized, format: 8.5 x 11 inches (.

21,6 x 28 cm)].

They are not returned, hence we advise the authors to keep a copy.

The title of the paper should be writing with capital letters.

The author's name has to apply in the middle of the line, near the title. References should be mentioned in the text by a number in square brackets and should be listed alphabetically.

Current address followed by e-mail address should apply at the end of the paper, after the references.

The paper should have at the beginning an up to a half-page abstract, followed by the key words.

All manuscripts are subject to anonymous review by two or more independent sources.

3

A FUNCTION IN THE NUMBER THEORY

Sunnarv In this paper I shall construct a function 0 having the following properties: (1 )

'7 n E Z

(2)

ry(n)

n ~ 0

is the smallest natural number with the

property (1).

We consider: N* = {1, 2, 3, Lemma~.

'7 k, P

=

i

p-1

=

< p - 1, j

Proof.

{O, 1, 2, 3,

... } and

... }.

under the shape:

=

=

N

1,

N*, P (p)

k = t.a In

e. , n,

..

I

~1,

(p)

> n 2 > •..

•

str~ng

(p)

(an

k is uniquely written

) neN'"

E

i

> n t > 0 and 1 < t

ni

,

ti

E

i

N,

•

cons~sts

increasing infinite natural numbers and

'7 n

(p)

fllhere a +- ••• +- t.a <.. n e. n

1 < tt ~ p,

1,e.-l ,

The

E

of

(p) a~,

=

=

1,Z,

(p)

- N* =

AMS (MOS)

=

(p)

1, a 2

=

(p)

(p)

u ( [an n€N*

1 + p,

(p)

a3

=

E

•

-1

= p

2

1 + P + P ,

(p)

,an+i ) n N*) where [an

e.

str~ctly

N*, P is fixed, a,

<

j

(P)

,an+l) n

subject classification (1980): 10A99

• a

(p) ,"1

'

N*.

.

.., (p)

(p)

a~2)

n [an+l'

=

0

(p)

because

Let

<

(p)

=

k €N*, N*

a~2

a~1)

u

=:! n..

N*)

n

n€N*

(p)

an

the shape k

- k is uniquely written

1

k

=

(p)

an

,

=

t

k

We note

E

a (p)

+ r,

,-

k€

N*:

unde~

(integer division theoren) .

+ r

k

1

,

n,

If r,

=

(p)

0,

as an

,

< k < an

(p)

1 i'1

- 1 - 1 < t, < P and Lemma

1 is proved.

If r, '* 0 - 3

r,

r

(p)

i

.n 2

.a

€

t

(p)

> r, - n, > n z ' r, ~ 0 and an

,

1 -

1 because we have t, < (an 1+ 1 -

1 -

The procedure continues similarly. number of steps t, we achieve r t

=

•

(p)

(p)

< P -

1 < t. <

r,)

an

< P. I

After a finite

0, as k

=

finite, k E N*

and k > r

1

= a

> r 2 > ..â&#x20AC;˘ > re.

only a finite

nu~ber

and bet-,.;een

a

and k there

lS

nu~~ers.

of distinct natural

Thus:

k

is uniquely written:

r is uniquely written:

=

k

.,. . =

-,

(p)

t 1a n

r

1

(p)

t 2a n 2

1,

1 <

1,

r t - 1 is uniquely written:

+ r e. and r t

e.-'

a

1,

j

=

1 < tt < p,

l,t-1,

t

=

=

t

because n t

a(P)

1 n

E

1

+

N*,

> 1.

(P)

Let k

E

N*,

k

=

t l ar,

0,

1 < tt ~ p,

-k is uniquely written under the shape k

wi th n l > n 2 > .â&#x20AC;˘. > ne. > 0; n t >

1,

+

1 < t2 < P -

< p -

< P -

tl

with

=

p-1

1 < t.

J

<

1.

=

1,

1 < t

j

e., e

> 1, n i ' t

< p -

1,

j

=

i

1,

€

N *,

e. -1,

1.

=

1,

e. , n 1 > n Z > ... >

r. e, >

1 < te, < P

I construct the function ryp' p

=

prime> 0, ry:: N* -

~

thus:

"in € N*

+ ...

NOTE .1.

) + ...

The function ryp is well defined for each

natural number.

Proof LEMMA ~.

'y'

k € N* - k is uniquely written as k

• with the conditions from Lemma .1

+

) and t,p

n,

= t~a~

LE~.A

}.

(p)

t,a '- n

= t.o ,.

(.

~

k c N*,

~

pEN, p

=

prime - k

with the conditions from

a......

+ an >

b

an

r --1

lb

(p)

ÂŁ -

ry

p

(k) =

n,

It is known that

+

Le~~a

=....'-, a n,

~

[~I b J

ai' b c N* where through [a] we have written the

J

integer side of the number a.

I shall prove that pIS powers

sum from the natural numbers which make up the result

n, is > k;

factors (t,p

n. t 1p

I

+

+ tt P

nt

n, t,p

+

>

P

...

+

+ >

nt

P

P

n, t,p

te. P

+ ... +

n

=

t,p

, -+ -1

n, ~.

.. .

~

t,P p

p

n1

n. p

n A dd in g - pI S po w er s su m is > t, (p 1 -1

I

+ .• . + pO) + .. . +

(p)

t.<.a n

(.

=

k .

THEOREM 1.

th e fu nc ti on n ' p = pr im e, de fi ne p d pr ev io us ly , ha s th e fo ll ow in g pr op er ti es : ( 1) "'f k € N* , (2 )

ry

( 1) •

p

(~is th e sm al le st nu m be r w it h th e pr op er ty

Pr oo f

(b y

(1 )

re su lt s fr om Lemm a 1.

(2 )

"'f k € N* , P > 2 - k = t, a

Lemma~)

(p)

n 1

is un iq ue ly w ri tt en , w he re :

9

a

=

i

l,e. , 1 < t.j < P - 1, j

n, - T7 p (k) =

t~p

=

1,e.-l

(p)

n i

=

,

1 <

E

N*,

p-l

t~

< p.

n.

nt + .•. + tt P

I

note:

I

Z

t 1p

=

Let us prove that z is the smallest natural number with the property (1).

I

suppose by the method of reductio

ad

absurdum that 3 YEN, Y < z : y!

=

'... .~lP Ie.

;

Y < z -

z - 1

pJ

=

1 -

(z -

1)!

=

YIp\(.

n,

=

n.J EN,

z-l --1

Y < z -

1; n, > n 2 > •..

t 1p

j

=

1, e.

n,-l t 1p

> n t > 1 and

"T"

•••

+ tt_,P

nt

_,-l + tt P

n t -l

-1

-1

- 1 as r l p

because p > 2 ,

1

= -

1

1.0

z-1

as P > 2

n~

I

>

1 -

I

n t _1 -n t -1 + tt_,P

2-1 T

=

1 as

-

•••

tt_,P

n t _,-n t -1

=

because

z-l

, Z-1

1n,

t 2p r O 1 = t,p +1.

n2

+

...

+ tt P

nt

-1

=

n,

1

O t,p ,

.

p

p

Because

a

< t 2p

n2

+

...

+ t{p

nt

-1 <

(p-1)

p

n2

+

+

11 nt,.,

-

+ (p-1) p

= p

< (p-1)

n z+1

1 <

-

(p-1 ).

1 <

n,

n,

-1 < P

- 1 < p

p-1

=

0

n.

I

p

n, t,p

z-l

+

=

=

n,+l

n,+l p

p

o

0 beca use:

n,+l

n,

- 1 < P

< t,p

n,+l

- 1 < P

acco rding

to a reaso ning simi lar to the prev ious one. whic h Addi ng - pIS powe rs sum in the natu ral numb ers make up the prod uct facto rs (z-l) ! is:

n,-l t,

(p

+ tt (p

+ ... ~ po) + .,. + tt,~, (p

n t -1

+ ... + po) - 1- n t

=

n t 路,-l

. + '"

+ po) _

k - nt, < k - 1 < k beca use

, ...

-~

nt > 1 -

(z -1)

. ... p , th~s "

~

(:

cont=adic~s

the supp ositi on

nade . is the smal lest natu ral

ryp(k)

n~~er

wi~h

~he

= ... p k . \1

cons truct a new func tion ry: z\{O} - N defin ed as

I

follo ws:

(

ry

=

(± 1)

a, P,

I

~

<

=

n

Pi

0,

€

p.J for i

II

as Ps with

... ~

€

=

j , a.1 > 1,

+ 1, Pi i

=

=

prim e,

1,s , ry (n)

=

!

= max

{ ry

o 1.

•

i=I,s

\

\

" NOTE

I.

(a)

~

ry

is well defin ed and defin ed over all.

€

Z , n

Proo f n

~

0 , n

~

± 1, n is uniq uely writ ten,

indep ende nt of the orde r of the facto rs, unde r the shap e of n

=

> 1

-

3~

a, P,

€

as .•• -Ps

with

€

=

_+ 1 wh ere PI'

=

(deco mpos e into prim e facto rs in Z ry

(n)

=

max l,S

{ry

Pi

(a.)} as s 1

=

prim e,

=

0

·i

;It

p j' a i >

fact oria l ring )).

finit e and ry P

(a.1)

€

N*

13

and:3

max i=l,s ry(n)

(b)

THEORE~~.

= o.

The function ry

previo~sly

defined has the

following properties: ( 1)

(ry(n»

(2)

ry (n)

I

•

-

-

n , "f n e: z\{O}

If

."1

is "the smallest natural number with this

property.

Proof ry(n) =

( a)

max

{ ry ;J

i

n

( a.) }, 1

a, =

e:

• P,

i=l, s (n

~+

1),

a (ry

(n

(a,) ) !

(as»!

ps

Supposing

=

,f

'

:.'~P1

= YIps

I

a s

max i=l,s

a. = :Ilp. 10) 10

T]

I

P

iO

(a i

)

d

e: N* and because

(Pi'P J') = 1, i

~ j

1

A

-~

j

(T7 p

,

=

1,s

(a ia »

1.0

(b)

n = -+ 1

(2)

(a) n

-

.

T7

(n) = 0; O! = 1, 1 = :,lE

1

-

a, n = E p,

as Ps

-

.

T7(n)

\,

1 = .I.n.

:::

r.1ax

77

i=l,s Let

max

( T7 p .

I

(a i) } = T7p

i=l,s T7

Pi

0

(a.)

I a

; -0

(a i

a

,

)

1 < i

<

S

is the smal lest natu ral numb er with the prop erty:

= '1,v

a. o. I a ~ I a

V YEN

Y < T7p

I

ia

y

-

(a.10 )

TJ

(b) numb er

o.1

~

.. M

€

•

P,

a.I

(a i

d-

a·

o. '0

•

1

0

is the smal lest natu ral numb er with the profe rt';.. _

n = ± 1 - TJ(n) = 0 and it is the smal lest natu ral 0 is the smal lest natu ral numb er with the 'prop erty

O! ::: M (± 1).

1 _::l

NOTE}. injec~ive,

The functions ryp are increasing, not

on ~* -

{pk

I

k = 1, 2,

... } they are surjective.

The function ry is increasing, it is not is surjective on Z \ CONSEQUENCE. n

=

N \

it

{1}.

Let n E N*, n > 4.

=

prime - ry(n)

CO}

injec~ive,

Then

n.

Proof It ~f1

n

=

prime and n > 5 - ry(n)

=

ry~(l)

=

n.

H{'l "

Let ry(n)

=

n and suppose by absurd that n

(a)

or n

=

ry(n)

=

max

a, p,

as p s with s > 2, a i

ryp

(a.

io

i=l,s

1

~

prime -

E

N *,

i

=

1, s

a ) <

contradicts the assumption; or

(b)

n

=

a, p,

a, with a, > 2 - ry (n)

=

I

ryp,(a,) < p,路a, < p,

=

because a, > 2 and n > 4 and it contradicts the hypothesis.

Apolication ~.

Find the smallest natural number with the property:

n

Solu-:ion

Let us calculate ry2(31) i we make the string

=

I,

3,

=

31

7,

15,

1'31 -

31,

ry2(31)

63,

=

) n €

ry2(1 31) 0

=

1 25 0

=

32.

(3)

1,

=

2

0

4,

13,

ry3(13)

)n

£

=

~_

.

40,

• • 'J

=

-+- l'ry3(1)

2'3 3 + 1'3'

=

=

54 + 3

57.

(7)

Let's calculate ry7(13); making the string (an

=

1,

=

1 7 2 + 5 7' =

84}

8,

. . . j13

57,

0

0

=

84 -

84!

=

...

Let's calculate ry](27) making the string (an =

~.

=

1 8 + 5'1 0

49 -+- 35

=

=

84 -

\l(± 231.327.713)

ry7(13)

=

l

o

ry(± 2 3"3 27 .7'3)

)n

ry7(8)

=

£

N-

=

-+- 5 '7 7 (1) 0

max (32,

57,

and 84 is the smalles'=.

number with this property. ~.

Which are the numbers with the factorial ending

1000 zeros?

Solution n = 10 1000 ,

Cry (n))! = M10 1000 and it is the smallest

number with this property.

ry ( 10 1000 )

=

T'] (

2 1000

0

5 1000 )

=

max' ( ry 2 (10 0 0), ry 5 (10 0 O)}

1 156 0

2'31

1•

= -I.

:J

lr

17 + 2-5 3 + 1-5 7

=

this property.

4005, 4005 is the smallest number with 4006, 4007, 4008, 4009 verify the property

but 4010 does not because 4010!

=

Florentin Smarandache University of Craiova

4009! 4010 has 1001 zeros.

17.11.1979

Nature Science Faculty

[Published on "An. Univ.

Timi~oara

",seria

~t.

Matematice,

Vol. XVIII, fasc. 1, pp. 79-88, 1980; See Mathematical Reviews: 83c : 10008.]

AN INFINITY OF UNSOLVED PROBLEMS CONCERNING A FUNCTION IN THE NUMBER THEORY

ยงl.

Abstract

w.

Sierpinski has

conference that i:

asse~ted

~anklnd

to an

las~ed

interna~ional

for ever and

nu~be=ei

~~e

unsolved problems, then in the long run all these unsolvei problems would be solved.

The purpose of our paper is that making an infinite number of unsolved problems to prove his supposition is net true.

Moreover, the author considers the unsolved

proble~s

proposed in this paper can never be all solved!

Every period of time has its unsolved problems which were not previously recommended until recent progress.

Number of new unsolved problems are exponentially increasihg in comparison with ancient unsolved ones which are solved at present.

Research into one unsolved problem may produce

many new interesting problems.

The reader is invited to

exhibit his works about them.

ยง2.

Introduction We have constructed (*) a function ry which

associa~es

to each non-null integer n the smallest positive integer such that m! is a multiple of n. standard form:

Thus, if n has the

~

19

with all o·I distinct primes, ~

all a j E N*, ry

C::.

1)

and E

=

+ I, then ry(n)

=

max

{ryP

i

l~i~r

(a i ) } , and

= o.

a E N*; then ryp(a) that b!

is the smallest positive integer b such

is a mUltiple of pa.

=

a~d

let p be a prine

Now, we define the ryp functions:

, k

=

I,

Constructing the sequence:

2,

p-l

(p)

we have ryp (a~

)

= pK,

for all prime p, and all k

=

Because any a E N* is uniquely written in the

=

a

and 1

~

t a

I,

fo~:

(p)

1 n 1

tj

~

P - 1 for j

=

0, I,

... , e - I, and 1 < te < p,

with all n i , ti from N, the author proved that

•e

=

§3.

nj

2: tiP i=1

Some Prooerties of the Function n Clearly, the function ry is even:

n E Z*.

2,

If n E N* we have:

ry(- n) = ry(n),

28 ry (n)

-1 (:.. ) ( :;

-:.. )

1

n

!

is

a::c.:

-,

<

<

~axinum

i: and only i: n is prlme or n =

~;

n 7'7 (n)

is nlnlmum if and only if n

= k!

n

Clearly

is not a periodical function.

ry

functions N* -

ryp

are increasing, not injective but on

I k = 1, 2, ... } they are surjective.

{pk

find that

ry

= o(n'1-,),

The function ('7')

m

~

Z~);

For p prlme, the

n e N*,

(=)

ry

= O(n).

e > 0, and ry

is aenerallv increasina on N*, that is:

rna e N*,

rna we have ry (m)

From (1) we

~

l'!'-o

(n)

ry

= ffio (n), such that for all (and generally decreasing on

it is not injective, but it is surjective on

Z \ { O} -

N\ { 1 } â&#x20AC;˘

The number n is called a barrier for a numbertheoretic function f(m)

if, for all m < n, m + fern)

Erdos and J. L. Selfridge). many barriers, with a < e

~

Does e 1?

a ma e N such that for all n - 1

ry

~

n (P.

(m) have infinitely

[No, because there is ~

rna we have

ry

(n -1) >

2

(ry is generally increasing), whence n - 1 + e ry (n - 1)

>

e > n + 1.] ~ n~2

1/17 (n)

is divergent, because 1/ry(n) > lin

~

; .

2

n 2

21 2

Let

Proo f:

2

=

T']

2 -+- 2

1

!

k-l time s

k time s (2)

am

=

2m

-

I, wher e m

=

n 2

;

2 k-2 time s

(2)

(1 + a,lI

then

T']

ยง4.

Glos sarv of SYmb ols and Notio ns

(

A-se quen ce:

T']2

=

an integ er sequ ence 1

=

(1)

T']2

~

so

a, < a 2 < '"

that no a j is the sum of dist inct

me~~ers

of the sequ ence othe r than a j (R. K. Guy) ; Aver age Orde r:

if fen)

is an arith meti cal func tion and

g(n) is any simp le func tion of n such that + g Cn)

f(l) + ... + fen) - gel) + we say that fen)

is

~f

the avera ge orde r

of g(n) ; numb er of posi tive divis ors of x; diffe renc e betw een two cons ecuti ve prim es:

co

Diri chle t Seri es:

a serie s of the form F(s) may be real or comp lex;

=

~

, s

22

Gene ratin g Func t.ion:

~

any funct .ion F(s)

=

~

~s

an Un(s)

n=l cons idere d as a gene ratin g func tion

an; the u n (s)

=

~ost:

usua l fa=::! of u.,(s)

e

J..

0:

is:

is a sequ ence

n

of posit .ive numb ers whic h incre ases stea dily to infin ity; Log x:

Napi erian loga rithm of x, to base e;

Norm al Orde r:

fen) has the norm al orde r F(n) appr oxim ately F(n) of n,

i .e.

• F(n) < fen)

( 2),

if fen)

is

for almo st all valu es

( Ii)

> 0,

€

< (1 + €).

(1 -

F(n)

E) •

for alnos t.

all valu es of n; "alm ost all" n mean s tha":. the numb ers less than n whic h do not poss ess the prop erty (2) Lips chitz Cond ition :

is

0

(x);

a func tion f veri fies the Lips chitz cond ition of orde r a € (0, 1J if (~)

a

=

k > 0:

If(x) -f(y)

I

~ k

Ix-yl a ; if

1, f is calle d a k Lips chitz -fun ct.io n;

if k < 1, f is calle d a cont racta nt func tion; Mul tipli cativ e Func tion:

a func tion f: N* - C for whic h f(l) and f(m • n) = f(m)

= p (x) :

• fen)

1;

larg est prim e facto r of x;

= 1,

wherl (m, nl

23

Uniformly Distributed:

a set of ooints in (a, b) distributed if every

is

unifor~ly

sub-inte~va:

0:

(a, b) cO;1tains its p=oper quota points; Incongruent Roots:

s-additive sequence:

two integers x, y which satisfy the congruence f(x)

- fey)

that x â&#x20AC;˘ y

m) ;

(~od

a sequence of the

=0

fo~:

a1

(~od ~)

=

and so

=

as

Queneau) ; sum of aliquot parts (divisors of n othe=

s (n) :

than n) of n; o(n) - n; kth iterate of s (n) ; s*(n) :

sum of unitary aliquot parts of n; least number of numbers not exceeding n, which must contain a k-term arithmetic progression; number of primes not exceeding x;

1T(X) :

a, b):

1T(X;

number of primes not exceeding x and congruent to a â&#x20AC;˘ modulo b;

o (n)

:

sum of divisors of n;

01

(n) ;

sum of k-th powers of divisors of n; k-th iterate of o(n) : 0* (n) :

sum of unitary divisors of n;

=

<pen) :

Eule~ls

totient function;

nu:nbe~s

not exceeding n and

k-th

ite~ate

ove~

the distinct

n~~be~ p~i::le

c: tc

:1;

of <p(n);

r:;; (n) :

n (n)

nU::lbe~

:

p~ime

diviso~s

cf :1;

of pri::le factors of n, counti:1g

repetitions; w (n) :

numbe~

LaJ :

floo~

of distinct prime factors of n; of a; greatest integer not greater

than a; (m,

g.c.d.

n):

(greatest common divisor) of m a:1c

n;

em,

loc.d.

n]:

If I: f(x)

(least

multiple) of m and n·

co~uon

"

modulus or absolute value of f; - g(x):

f(x)/g(x) -

1 as x -

00; f is asymptotic t

g; f(x)

=0

(g(x)):

f(x) f(x)

= O(g(X))}

«g(x)

;

f(x)/g(x) -

(X) :

as x - 00;

there is a constant c such that If(x) I < < cg(x),

r

a

for any x;

Euler's function of first case function); r dt.

E

R*. -

We have r(x+l)

R,

=

N*, rex) = (x - l)!

(ga~~a

rex)

x rex).

If X£

25

f3(x) :

Euler's function of second degree (beta function); f3 : R*. x R*. - R, 1

f3 (u, v) = r

(u) r

(v)jr (u

v)

J

=

t:..i

o ;

o

路(1 - t)VO' dt; Mobius' function; J.i.. : N - N

J.i.. (x) :

J.i..

(n)

= (-

=

J.i..(1)

1;

1)~ if n is the product of

k > 1 distinct

pri~es;

J.i..

(n)

=

0 in all

other cases;

a (x)

Tchebycheff a-function; a

:

a (x)

=

R.. -

R,

2: log P

where the summation is taken over all primes p not exceeding x; 'P'(x):

Tchebycheff's 'P'-function; 'P' (x)

=

2:

=

A (n), with

n~x

A (n)

=

log p, if n is an integer power of the prime p; 0, in all other cases.

This glossary can be continued with OTHER (ARITHMETICAL) FUNCTIONS.

搂5.

yes,

, General Unsolved Problems Concerning the Function T7

(1)

Is there a closed expression for ry(n)?

(2)

Is there a good asymptotic expression for

find it.)

~(n)?

(If

26

(3)

Fo~

a fixed non-null

(Pa~ticula~ly

m?

t~ivial:

(4 )

when n

=

we find n

=

w, does ry(n) divide n-

lntege~

Of course,

1.)

k!, or n is a

s~~a~ef~ee,

Is ry an algebraic function?

max Card {n

z* I

£

(3)

p

£

(If no,

f

non-null.

is there the max Card {n

null, fen, (5)

=

ry(n))

~

etc. is

t~e~e

t~e

More generally we

€

z* I

Is f

(3)

fc~

a f-function?

ry

R [x, y],

€

f non-

0 for all these n}?)

Let A be a set of consecutive integers from N*.

Find max Card A for which ry is monotonous.

A

0 it is

g is a f-function if f(x, g(x)) = 0

all x, and f €"R [x, y], no,

=

R [x, y], p non-null polync::.ial,

with pen, ry(n)) = 0 for all these n}?) introduce the notion:

for m

5, because for A = {1, 2, 3, 4, 5}

ry

For example, Card

is 0, 2, 3, 4, 5,

respectively.

(6) €

A number is called an n-algebraic number of decree n

N* if it is a root of the polynomial (p )

p 77 ( x)

=

ry (n)

~ + ry (n -

1 ) ~-1 + ... + ry ( 1 ) x'

= o.

An n-algebraic field :'-1 is the aggregate of all numbers A (u) =--

8(u) where u is a given ry-algebraic number, and A(u), 8(u) are polynomials in u of the form (p) with B(u) (7)

Are the points Pn

=

~

o.

Study \1.

ry(n)/n uniforillly distributed in

the interval (0, 1)?

(8) is ry(n), n

Is Q.0234537465114 ... , where the sequence of ~

1, an

i~rational

number?

digi~s

27

* Is it possible to rep=esent all integer n under the

= ... =

(9 )

the integers k, a l

,

... ,

F-.,....... l ..

_~_

a., i7

, where

(a ie )

a ie , and the signs are conveniently

chosen?

17 (a 1 ) a1 +

(10)

But as n

=

~

(11)

But as n

=

+ a1

?

17 ( a z)

+

?

* Find the smallest k for which: of the numbers 17(n), 17(n + 1),

... ,

(~)

17(n

n e N* at least one ~

k - 1) is:

(12) A perfect square.

(13) A divisor of kn.

(14) A mUltiple of a fixed nonzero integer p.

(15) A factorial of a positive integer.

* (16)

Find a general form of the continued fraction

expansion of 17(n)/n, for all n

~

2.

(17) Are there integers m, n, p, q, with m p - q,

for which:

17(m) + 17(m + 1) +

~

n or

+ 17(m + p)

=

17(n)

+ 17(n + 1) + ... + 17(n + q)?

(18) Are there integers m, n, p, k with m such that:

(19)

How many primes have the form:

* nand p

> 0,

23

ry(n~l)

ry(n)

..•

1')(n ~ k)

for a fixed integer k?

=

Fer exahlple:

ry(2)

1')(3)

(20)

P::-ove that ry(xn)

=

23, 17(5) 17(6) ~

53 are p::-l:::es.

ry(y")

Look,

integer solutions, fo::- any n > 1. solution (5, 7, 2048) when n More generally:

theore::t. )

=

(On

3.

the

n(zn)

=

has an for

exa~ple,

Fe~at's

diophan~ine

infi~i~y

at

of ~he

last

equation

k L

i=l s

=

1') (x)

m

t

17 (y.) has an infinite number of solutions.

L

j=l

J

(21) Are there m, n, k non-null positive integers, m • 1 • n,

for which 17(m • n)

mk

•

17(n)?

Clearly, 17 is not

to degree k.

ho~ogenous

(22)

=

Is it possible to find two distinct numbers k, n 17(n K) be an integer?

for which log

(The base is 17 (kn) . )

17 (kn) (23) X1)(1)

=

Let the congruence be:

0 (mod m).

C

(24) We know that eX co

L x,,(n} / n=l

n!,

L x" / n=l

x1)(n}

+ ... + c 1 •

How many incongruent roots has h", for

some given constant integers n, c"

co

n

••• I

=

c ., n'

Calculate

.

17 (n)! and eventually some of their

properties.

(25)

Find the average order of 17(n)

(26)

Find some u n (s)

for which F(s)

be a generating

function of 17(n), and F(s) have at all a simple

fo~.

29

co

Particularly, calculate Dirichlet series F(s)

=

ry(n)/n S ,

~

n=l with s c R (or seC) .

Does ry(n) have a normal order?

(27)

(28) We knqw that Euler's constant is

y

=

1

1

lim n-aJ

+

1

n - log n)

2

n

Is lim n-aJ

1

+

1

~

1/1')

(k) - log

1')

(n) i

j

k=2

a constant?

If yes,

find it.

(29)

Is there an m for which

such that the numbers ai' a 2 ,

••• ,

1').1

(m)

a~

=

... ,

(a 1 , a 2 ,

a pc')

can constitute a

matrix of p rows and q columns with the sum of elements on each row and each column is constant?

particularly when the

matrix is square.

* (s)

(30) Let

{x n

possible to have

}n>1

(s)

1')(xn

be a s-additive sequence. ,

•

n ". m?

(31) Does

1')

(32) Is

1')

a k-Lipschitz Condition?

(33) Is

1')

a contractant function?

But

verify a Lipschitz Condition?

Is it

=

(5 )

ry(x n )?

(34)

Is it possible to construct an A-sequence a.,

a,., such that rJ(a 1 ) , for

exa~ple

2,

... , ry(a) be an A-secruence, n

3, 7,

~

31,

~oo?

•••

I

Yes,

Find such an infinite sequence.

* Find the

greates~

'.;::

n such that:

~~

a"

... , an

constitute a p-sequence then ry(a,) , ... , ry(a.,) p-sequence, too; where a p-sequence

consti~ute

a

~eans:

(35) Arithmetical progression.

(36) Geometrical progression.

(37) A complete system of modulo n residues.

Remark:

let p be a prime, and p, p2,

geometrical progression, then ry(pi)

=

... , pP a

ip, i E {1, 2,

... ,

p}, constitute an arithmetical progression of length p.

In

this case n - co.

(38) Let's use the sequence an

=

ry(n),

a recurring relation of the form an

=

f(a n_"

n

~

1.

Is there

a n-2, ... ) for

any n?

(39) Are there blocks of consecutive composite numbers m + 1,

... , m + n such that ry(m + 1),

composite numbers, too?

••• 1

ry(m -+- n) be

Find the greatest n.

(40) Find the number of partitions of n as sum of

rJ(~),

2 < m < n.

MORE UNSOLVED GENERAL PROBLEMS

CONCE~~ING

THE FUNCTION

rJ

31

§6.

Unsolved Problems Concernina the Function

and Using

n

the Number Seauences 41-2065)

... ,

Are there non-null and non-prime integers a 1 ,

... ,

an in the relation P, so that ry(a,), ry(a z ) ,

ry(a) be in the relation R? t1 property.

Find the greatest n with this

(Of course, all a j are distinct.)

eac~

Where

P,

R can represent one of the following number sequences:

N is abundant if a(a) > 2 a.

(1)

Abundant numbers; a

(2)

Almost perfect numbers; a

(3)

&~icable

€

€

=

N, a(a)

2a - 1.

=

numbers; in this case we take n

are called amicable if a

~

band a(a)

=

a(b)

=

a + b.

Augmented amicable numbers; in this case n

(4)

b are called augmented amicable if a(a) = a(b)

2; a, b

=

=

2; a,

a + b - 1

(Walter E. Beck and Rudolph M. Najar).

Bell numbers:

(5)

bn

=

n ~

Sen, k), where Sen, k) are

k=l

stirling numbers of second case.

(6)

Bernoulli numbers (Jacques 1st):

the

Bn ,

coefficients of the development in integer sequence of

t

t

= 1 -

for 0 <

I tl

B, + - tZ 2 2~

< 2 rr;

+ (_ 1) n- 1 4

~

t

(2n)

(here we always take L1/BJ).

~

Zn

+

... ,

32

1 Catalan

(7 )

=

nu~ers:

2n-2\ n-l I

n

n .2: 2.

Ca=michael numbers; an odd composite

(8)

pseudopri~e

which is a pri~e

to base b for every b

to a, is called a Carmichael

(9)

Congruent nuillbers; let n

b, c; we must have a

=b

n~nber

a,

rela~ivelv

n~~~er.

=

3, and the

n~~~ers

a,

(mod c) .

(10) Cullen numbers: (11) C1-sequence of integers: the author introduced a sequence a" ('i)

i

€

N*,

a 2,

... so that:

(3)

j, k

€

~

j

N*,

~

i

k .. j,

(12) C2-sequence of integers; the author defined other sequence a l ('i)

i

€

N*,

,

a 2,

... so that:

(3)

j, k

€

i

N*,

~

(13) Deficient numbers; a (14) Euler numbers: expansion of sec x

=

• k • i,

j

a(a) < 2a.

€ N*,

the coefficients En in the

l: En xn/n!: here we will take

n.2:0

(15) Fermat numbers:

.... =

:n

(16) Fibonacci numbers:

22

n

=

1=

.~1

+ 1'

f2

=

IEn I .

n.2: 0 .

1, fn

=

f n· 1 +

0: J.

n- 2 I

n > 3• (17) Genocchi numbers: Bernoulli numbers: always Gn

Gn €

=

Z.

2

(2

2n

-

1) Bn , where Bn are

33

(18) Harmonic mean; in this case every mesner of the sequence is the harmonic mean of the preceding members.

(19)

Harwonic nunbers; a number n is called

ha~~onic

~;

the harmonic mean of all divisors of n is an integer (C. Pomerance) .

(20) Heteromeous numbers:

h :1 = n

(n + 1),

n

E

N*.

(21) K-hyperperfect numbers; a is k-hyperperfect if a

=

1 + L di

,

where the numeration is taken over all proper

divisors, 1 < d i < a, or k a(a)

=

(k + 1) a

~

k - 1 (Daniel

Minoli and Robert Bear).

(22) Kurepa numbers: + (n

-

!n

=

O!

~

1! + 2!

1) !

(23) Lucas numbers:

L,

= 1, T._ L -- Ln-' -z = 3 ' n

n > 3.

(24) Lucky numbers:

from the natural numbers strike

out all even numbers, leaving the odd numbers; apart from 1, the first remaining number is 3; strike out every third member in the new sequence; the next member remaining is 7; strike out every seventh member in this sequence; next 9 remains; etc.

(v. Gardiner, R: Lazarus, N. Metropolis, S.

Ulam) (25) Mersenne numbers:

M

P

=

2P -

1.

(26) m-perfect numbers; a is m-perfect if

a~(a)

= 2a

(D. Bode).

(27) Multiply perfect (or k-fold perfect) numbers; a is k-fo~d

perfect if a(a)

=

k a.

=

(28)

Pe~fect

(29)

Polygonal numbers (represented on the perimeter

a polygon): (30)

Ie

Pn

numbers; a is perfect if a(a)

= k

2a.

0:

(n - 1).

Polygonal numbers (represented on the closed (k-2)

(k-4) n

surface of a polygon) : 2

(31)

Primitive abundant numbers; a is primitive

abundant if it is abundant, but "none of its proper divisors are.

(32)

Primitive pseudoperfect numbers; a is primitive

pseudoperfect if it is pseudoperfect, but none of its proper divisors are.

(33) Pseudoperfect numbers; a is pseudoperfect if it is equal to the sum of some of its proper divisors (w.

Sierpinski) .

(34) Pseudoprime numbers to base b; a is pseudoprime to base b if a is an odd composite number for which b a - 1 == 1 (mod a)

(C. Pomerance, J. L. Selfridge, S. Wagstaff).

(35) Pyramidal numbers:

~n

=

1

n (n + 1)

(n + 2),

6

n â&#x201A;Ź N*.

(36)

Pythagorian numbers; let n

=

3 and a, b, c be

integers; then it must have the relation:

a2

=

b2 + c2 â&#x20AC;˘

35

(37) Quadratic residues of a fixed

pri~e

p:

=

nonzero numbers r for which the congruence r

the

XZ (mod p)

has solutions.

(38)

=

a(a)

Quasi perfect numbers; a is quasi perfect

i;

2 a + 1.

(39)

Reduced amicable numbers; we take n

integers. a, b for which a(a)

=

=

a(b)

=

2; tNO

a + b + 1 are called

reduced amicable numbers (Walter E. Beck and Rudolph M. Najar) .

(40) Stirling numbers of first case: sen, k) x

(x -

s(O, 0)

is the coefficient of x K from the development

(x - n + 1).

1)

(41) Stirling numbers of second case:

and Sen, k) X Ck )

=

X

= 1, and

S(O, 0)

=

1,

is the coefficient of the polynom

(x -

(x - k + 1), 1

1)

~

k

~

n, from the

development (which is uniquely written) : n L

xn =

S

(n, k)

X Ck )

â&#x20AC;˘

k=l (42) Superperfect numbers; a is superperfect if aZ(a)

=

2 a

(D. Suryanarayana).

(43) Untouchable numbers; a is untouchable if sex)

= 1

has no solution (Jack Alanen) .

(44) U-numbers:

starting from arbitrary u, and Uz .

continues with those numbers which can be expressed in just

35 su~

one Nay as the

of two distinct

ea~lier

illembers of

sequence (5. M. Ulam).

Weird numbers; a is called weird if it is

(45)

a~u~ca~~

but not pseudoperfect (S. J. Benkoski).

MORE NUMBER SEQUENCES

* The unsolved problem No. 41 is obtained by taking P

=

and R

( 1)

=

( 1) •

The unsolved problem No. 42 is obtained by taking P

=

( 1)

I

R

=

= =

( 2) •

The unsolved problem No. 2065 is obtained by taking P = (45)

and R = (45).

OTHER UNSOLVED PROBLEMS CONCERNING THE FUNCTION ry AND USING NUMBER SEQUENCES

* §7.

Unsolved Diophantine Equations Concernina the Function n Let 0 < k

~

1 be a rational number.

Does the • diophantine equation ry(n)/n = k always have solutions? Find 2066)

all k so that this equation has an infinite number of solutions. h

=

I, 2,

(For example ...

I

I

if k

=

l/r ,re: N*

all Pa+h are primes I

such that Pa+l > r.)

I

then n

=

rpa+h'

and a is a chosen index

37 2067) Let {a} r1

an.'

=

ary(n) -+- 'I (an)'

n,

be a sequence, a o

n~O

=

1, a,

=

2, and

Are there infinitely many pairs (m, n),

=

for which a :n

(For exawple:

a n?

the equation 'lex)

2068) Conjecture:

=

r](x -+- 1) has no

solution.

* Let m, n be fixed integers.

Solve the diophantine

equations:

2069) r](m x + n) = x.

=

2070) r](m x + n)

m + n x.

2071) r](m x + n) = x!

2072) 'I

(X''ll)

xn.

=

2073) 'I (x) In = 'I (xn) .

=

20.7 4 ) 'I (m x + n) 2075)

T)

=

(x) + y

x + 'Icy) , x and yare not primes.

=

2076) 'I (x) + 'Icy) primes.

r](x)Y.

r](x + y), x and y are not twin

(Generally, 'I is not additive.)

2077)

T)

(x

+

= 'I (x)· 'I (y) •

y)

(Generally, 'I is not an

exponential function.)

2 0 7 8)

'I (xy)

=

T)

(x)

(Generally, 'lis not a

'I (y) •

multiplicative function.)

2079) r](m x + n)

=

x Y•

2080) 'lex) y

=

x r](Y), x and yare not primes.

2081) r](x)/y

=

x/r](y) , x and yare not primes.

(Particularly when y rational number.)

=

2 Ie , k

€

N,

•

~.e.,

r](x)/2 Ie

•

~s

a dyadic

38

...

~

= x:)(Y) , x and yare not prines.

2082)

1') (x)

2084)

1') (x Y) - 1')

(ZW)

=

I, with Y .. 1 .. w.

(On Catalan's

problem. ) 2085)

1') (x Y)

= m, Y

2086)

1') (xx)

=

2087 ) 1') (xY)

1') (x)

2089)

1')(m x)

2.

yY.

(A trivial solution:

x

= y =

= yx.

(A trivial solution:

x

= y

=

3. )

(An example:

= y!

2088)

~

= m 1') (x) , m

2090 ) m17 (X) + 1') (x) n

x

=

9, Y

2.)

= 2. )

> 2.

= mn.

2091)

1') (x 2 ) 1m

2092)

y, 1') (Xi +

+ Xr

2093)

1') (Xi! +

+ Xr !)

2094)

( x, y)

= (1') (x) , 1') (y) ) , x and y are not primes.

2095)

[ x, yJ

= [ 1') (x) , 1') (y) J , x and y are not primes.

+

1') (y2)/n Yr

= )

1.

= 1') (Xi)

y,

= 1') ( Xi) !

+ +

... + 1') (x r ) Yr ... + 1') (X r ) !

OTHER UNSOLVED DIOPHANTINE EQUATIONS CONCERNING THE FUNCTION

~ONLY

* ยง8.

Unsolved Diophantine Equations Concerning the Function n in Correlation'with Other Functions Let m, n be fixed integers.

equations: 1') (x)

2096-2102)

1')(x)m

=

1') (x) +

= d(m x

d(xn)

y=

x + dey)

+ n)

Solve the diophantine

39

1'] (x)

• y = x

• dey)

1'](x)/y = d(y)/x 1'] (x) y = xd(y)

2103-2221)

Same equations as before, but we substitute

the function d(x) rr ( x),

rr ~ x; m,

n(x),

w(x)

n),

with d x ' Ok (

x),

p(x), (x),

OK

0

s(x),

Sk(X),

* (x),

cp (x),

s*(x), cp k (x),

=(x),

~ ( x) ,

respectively.

2222)

1'](s(x, y))

= s(1'](X),

2223)

1'](S(x, y))

= S(1'](x),

2224 )

1'] (L xj)

= Lr

( x)j

?22S)

1'] (Lx -

yj)

= L.B (x,

2226)

.B(1'](Lxj),

2227)

1'] (L.B(x, y)j)

2228)

/-L(1'](x))

2229)

1'] (x)

2230)

1'] (x)

2231)

17(m

2232)

17(m

2233)

17 (m x + n)

y)

1'](y)).

1'](y)).

• y)j.

=.B(x,

1'](Lyj))·

= L.B(1'](Lxj),

1'](Lyj))j·

=/-L(<P(x)).

= L8(x)j.

= L1jJ(x)J. n x + n) = Ax = m x + n) = Ax· =

(

~

x(x -

. ..

(x -

x!

1

=

2234)

17 (.m x + n)

2235)

17 (m x + n)

= (~ 1 = px =

2236)

17(m x + n)

= L l/BJ

2237)

17(m x + n)

=

Gx •

1)

n!(x-n)!

the x-th prime.

.

n + 1) .

40 2238)

T7(:n x + n) = K;( =

2239)

T7(m x + n) = k:nx·

2240) T7(m X + n)

=

n

s (m, X) .

2241 ) 17(m X + n) = sex, n) .

2242)

T7(m X + n) = SCm, x) .

2243) 17(m X + n) = Sex, n) .

2244)

T7 (m X + n) =

7r x •

2245) 17(m X + n) = b x•

= I Ex I = X

2246)

ry(m X + n)

2247)

17 (m X + n)

2248)

T7 (x) == 17(Y)

2249)

T7(XY) - X (mod Y) .

2250) 17 (x)

(mod m) .

(x + m) + T7 (y)

2251) 17(m X + n) 2252)

I

17 (m X + n)

2253) 17(m X + n)

= = =

(y +

m)

=

17(Z)

(z + m) .

f x• Fx· Y1x·

2254 ) T7 (m X + n)

= ex·

2255) 17 (m X + n)

=

ex·

2256) 17 (m X + n)

=

hx •

2257) 17 (m X + n)

=

~.

More unsolved diophantine equations concerning the function 17 in correlation with other functions.

*

~

,

... ..1..

§9.

Unsolved Dioohantine Equations Concernincr the Function n in Comoosition with other Functions 2258)

ry

(d (x))

2259-2275)

Sa~e

=

d(ry(x)), x is not prime.

equations as

the function d(x) with d x ' p(x),

t~is,

but we substitute

... , w(x)

respectively.

More unsolved diophantine equations concerning the function ry in composition with other functions. example:

ry(7r(4(x)))

=

(For

q>(ry(rr(x))), etc.)

* §10. Unsolved Dioohantine Inecruations Concernina the Function n Let m, n be fixed integers.

Solve the following

diophantine inequalities: 2276)

ry (x) > ry (y) .

2277)

is 0 < (xjry(x)} < {ry(x)jx} infinitely often?

where {a} is the fractional part of a.

2278)

ry(m x + n) < d(x).

2279-2300) Same (or similar)

inequations as this, but

we substitute the function d(x) with Px' p(x),

•••

I

w (x) ,

rex), (3(x, x), }.L(x) , Sex), 'P'(X) , respectively . .

---------------------------------------------------------More unsolved diophantine inequations concerning the function ry in correlation (or composition, etc.) with other functions.

(For example:

S(ry(LxJ)) < ry(LS(x)J), etc.)

*

§11. Arithmetic Functions constructed bv Means of the Function n UNSOLVED PROBLEMS CONCERNING THESE NEW FUNCTIONS

r.

The function S17 : N* 2301) Is

!: x~2

S17 (x)

-1

-

N, S1) (x)

=

T'/ (n) .

i:

O<n$.x

a convergent series?

2302) Find the smallest k for which (S \

"

0

•••

~

S ) ~

(n)

>

k times

> n,

for m, n fixed integers.

2303-4602) Study S".

The same (or similar questions

for S1) as for T'/.

II.

The function C" : N* - Q, C1)(X) + T'/(x))

+

=

1 (T'/(1)

+ T'/(2)

x

(sum of Cesaro concerning the function

T'/) .

4603) Is

i:

C1) (x) -1 a convergent series?

x~l

4604) Find the smallest k for which (C1)o ... oC 17 ) (m) > . , k times >

n, for

!Il,

n fixed integers.

4605)-6904) Study C1). for C1) as for T'/.

The same (or similar) questions

III.

The function Ery

k () 2:

N* - N,

r,c'() (x), where

k=l T'](1)

=

T']

and

T'](k)

=

T']O

•••

smallest integer k for which 6905)

Is

2: x~2

6906)

E'7 (x)

-1

OT']

T'](k.1)

of k tines, ar.d k() (x)

=

1.S

the

(x).

T']«)

a convergent series?

Find the smallest x for which E'7 (x) > m, where

m is a fixed integer.

6907-9206) Study E. ry

for S1) as for IV.

The same (or similar) questions

T'].

The function Fry

N\{O,

I} -

N,

Fry(x)

=

2:

T']p

(x).

O<p~x

p prime 9207)

Is

2: x~2

Fry (x)

-1

a convergent series?

9208-11507) Study the function

The same (or

similar questions for F'7 as for ry.

V.

The function a ry

!3(n)

=

N* - N, ary (x)

=

x 2:

!3 (n), where

n=l

0, if ry(n) is even; 1, if ry(n) is old.

11508) Let n

€

N*.

Find the smallest k for which

k times 11509-13808) Study ary.

for a1) as for

T'] •.

The same (or similar) questions

VI.

The

func~ion

mry : N* - N, mry (j) = a j , 1

integ ers, and mry Cn +

r

13809 ) Is

...,,, ~"f

= min. ~

1)

(x) -1

I

a

j

n,

<

.)1,

r;(a.I + a rlI

fixe~

etc.

. . convergen~ ser~e s_?

x~l

1381 0-161 09) Stud y m. ry

~

The same (or S -i-il~ -'cu< ~n... ~- ---J ""! =ls-i "--"-' .. ~

for mry as for r;.

VII.

The func tion .'vi ry

N* - N.

integ er sequ ence a 1 ,

A given finit e posi tive

an is succ essiv ely exten ded

••• I

by: ~~ry

(n +

:,1" (i)=

1)

aj

,

1611 0) Is

= maxi {r;(a i 1 ~ j

+ an-i) }' etc.

~ n.

i:

a conv ergen t serie s?

x~l

1611 1-184 10) Stud y Mry'

ques tions for VIII .

~lry

as for r;.

-1

-1

The func tion ~'Imin wher e '1- 1 (x) r;

-1

32

=

(6)

.2

3

=

{24, 24

N\{l} - N, r;min (x)

(a

.

€

I

N

3 , 2t.

-1

}

The same (or simi lar)

when ce r;min (6)

I

r;(a)

=

3 2, 3 2 , 3 2

=

9.

{'1- 1 (x)},

For exam ple

X}.

.

= min

. 2,

32

-1

1841 1) Find the smal lest k for whic h \

r;min

0

.

...

k time s -1

18412 -2 0711) Stud y r;min'

ques tions for

-1

T'J

min as for r;.

The same (or simi lar)

22 ,

-1

0

'1 min

IX.

-,

-1

The function T] card

N -

nu~ber

where Card A means the

20712)

T] card (X) == Card {T]-' (x)}

N,

I

of elements of the set A.

Find the smallest k for which

-,

-,

T] card 0 â&#x20AC;˘â&#x20AC;˘. 0 T] card

I

)

(rn)

> n,

for rn, n fixed integers.

k times

.,

T]~~.

20713-23012) Study

-,

The same (or similar)

questions for T] car ..... as for T].

X.

The function dry : N* 0:.1)

Le t dry

(1)

where d17

( x) ==

(Ie)

I dry

N, dry (x)

(x + 1 ) - dry

= (Ie)

1T](x + 1) - T](x) ( x)

I,

for all kEN * ,

(x) == dry (x).

23013) Conjecture:

(Ie)

dry

(1)

=

1 or 0, for all k

~

(This reminds us of Gillreath's conjecture on primes.) example:

I.

2.

For

,

~

... 0

T) ( 1)

=

0

2 T) ( 2 ) = 2

1 1

T) ( 3 ) = 3

1

0 1

T) ( 4 )

=

0

4 1

T) ( 5 ) T) ( 6 )

= =

1

0

5

1 1

2

1

T)(7) = 7 3

T)(S) = 4

1

1

2 T)(9) = 6 T)(10)= 5

5

T)(II)=11

1 7

T)(12)= 4

3 4 2

T)(15)= 5

3

1 11

1 10

0 11

T)(IS)= 6 13 T)(19)=19

1

1

0 1

1 0

1

1 1

1

1 2

7

2

0

1

0

0

2 0

8

2

1

1 0

0

1 1

0 1

1 1

5

9

10

T)(17)=17

4 6

0 1

0

0

0

1

1 0

1

1 0

1

0

1

1

1

0

1

1

1

1

2

2

0

1

2

2

2

2

2

2

1

T)(16)= 6

0 0

1

6

T)(14)= 7

3 3

0

1

9

T)(IJ)=13

2

0

1

1 2

0

2

0 1

0

0

1

3

1

2

0

0

1 1

0

2

1

3 3

2 2

1

0

4

6

3

4

0 1

0

0

0

1 1

2

3

4

4

2

3

4

0 1

1

1

0

1 1

0

0

1 1

1 1

0

0 0

0

1

0 1

0

1

4

1 1

0

2

3

0 1

7

1 1

1 14

T)(20)= 5

23014-25313) Study â&#x20AC;˘ quest~ons

XI.

<10

for d ~

The function

<I()

d~

The same (or similar)

as for T). w~

: N*

-

N,

w~

( x)

is the number of m, Hence, w~ ( x) >

with 0 < m < x, so that T) (m) divide x. > w (x) , and we have equality if x

=

-

1 or x is a prime.

47

Find the smallest k for which (w, a ... own)

. 25314)

...

(x) =

(

k times

= 0, for a fixed integer x.

,

25315-27614) Study w .

The same (or similar) questions

,

for w as for 7'].

XII. The function :,[1'/ : N* -

N, :.11'/ (x)

with 0 < m < xx, so that 7'](m) " / (3) examp 1 e ."1'

=

Card {1, 3,

is a prime, :"11'/ (p) 2

~

i

~

=

is the number of m,

is a multiple of x.

6,

12, 27}

9,

Card {1, a z '

... , a r

6.

If P

then all ai'

r, are multiples of p.

27615) Let m, n be integer numbers. k for which

},

=

For

(~,

a ... a M1'/)

'----

~

(m)

Find the smallest

n.

:,

k times The same (or similar questions

27616-29915) Study for路

:,1 7) as for 7'].

The function a7)

XIII.

N* -

N, a7) (x)

=

!:

7'](d).

dlx d>O

For example a7)(18) + 7'](18)

=

=

20, a1'/(9)

7'](1) + 7'](2) + 7'](3) + 7'](6) + 7'](9)+

=

9.

29916) Are there an infinity of nonprimes n so that a7)(n)

=

n ?

29917-32216) Study a7). for a1'/ as for 7'] ..

The same (or similar) questions

XIV.

The

func~ion ~ ry

~~(x) '/

: N - N,

nunbers n so that ry(n)

~

x.

is tte nunber of If

p~

<

pz < ... < P'( <

<

:l

< Pk.l

is the pri:aes sequence, and for all i = a.+l = 1, 2, ... , divides n! but -o.I does 1

not divide n!,

1Try

(n)

=

th~n:

(a, + 1)

32217-34516) study question for XV.

1Try

as for

The function m"'ry

1Try'

The same (or similar)

ry.

N* - N, <!lry (x) is the number of m,

with 0 < m < x, having the property 34517) Is

(ry(m),

=

x)

1.

always true that <!lry (x) < <!l (x)?

34518) Find x for which <!lry (x) > <!l(x).

34519) Find the smallest k so that (<!lry

0

•••

a

~

!9ry)

,

(x)

=

k times

= 1, for a fixed integer x.

34520-36819) study <!lry' for <!lry as for

The same (or similar) questions

1'].

More unsolved problems concerning these 15 functions.

More new (arithmetic) functions constructed by means of the function

1'],

and new unsolved problems concerning

them.

------------------------------------------------------------

q9 seq~ences

We can continue these recurring

36820 - co.

of unsolved problems in number theory to infinity. ~ore

construct an infinity of functions S17'

~17

••• ,

,-ne Using ....

new functions:

f ~ Z'

construct the functions

... , fln 1 (by varied cOwbinations bet!..;een S17' C17 , for example:

(x)

N* - N for all i

(i)

~

o<n.:s,x

S17

= 0, 1, 2,

4'17;

~*,

for all x E (0)

=

... , where S17

S17'

Or:

x

1 .

=

=

••• ,

~

S17

(n),

x n=l

SC ll

N* - Q, SC 17 being a conbination

between Sll and Cll ; etc.); analogously by means of the functions fll'

fl

f 1Z '

... , fZn

Z

etc.

continues to infinity.

n 1

we construct the functions

The method to obtain new functions For each function we have at least

2300 unsolved problems, and we have an infinity of thus

functions.

The method can be represented in the following

way: produces ••• ,

fll'

· .. , · .. ,

f'n .

i=

"'1 n

... ,

1

- - - - - - . f31'

•••

I

... ,

· .. ,

*

Othe~

ยง 12.

~ecu~ring

methods to wake new unsolved

preble~s.

Conclusion With this

pape~

construct infinitely nUlllber theory:

the author wants to prove that we ~any

unsolved

p~oble~s,

you "rock and roll l1 the

create interesting scenarios!

ca~

especially in until yeu

nu~:Oers

Some problems in this paper

could effect the subsequent development of mathematics.

The world is in a general crisis.

Do the unsolved

problems really constitnte a mathematical crisis, or contrary to that, do their absence lead to an intellectual stagnation?

Mankind will always have problems to solve,

they even must again solve previously solved problems(!)

For example, this paper shows that people will be more and more overwhelmed by (open) unsolved problems.

[It is easier

to ask than to answer.]

Here, there are proposed (un) solved problems which are enough for ever!!

Suppose you solve an infinite

nu~er

of

problems, there will always be an infinity of problems remaining.

Do not assume those proposals are trivial and

non-important, rather, they are very substantial.

171

§13. References (books and pa.pen which ia.ve inspired ,he a.uthor)

(I} Amouz Gabriel,

Arith~tiqu~

graphiquo. Introduction

a I'el:ldo

dl!3 fonctions

arithmet:qu~,

Gauthier.· Villar., Par'-3, 1906.

[2] Blanchard A., hitiation a la tluiorie

anaiitiq"~e d~s

nomaro. primien, Dunod. Pa,.....s, 1969.

;'3} Borovitch Z.J. and Shafa",-c:tch !.It, Number Theo,...}, Academic ?,.ess. 'vew Yor.i:. 1966. i~j

BouuierAla:.n et: George Jficiul (sous fa des

.Wathemat;qTJ.~s. Pross~s

rli~c!ion

de FrancolS Lt! Llonnals;,

DictioTlair~

UniversitalT'!S de Franc!, Par'..3, 1979.

(5} Carmichael R. D .. Theo,...; of Numbers, .Hathematicai 'vfonogra.ahs. So. 11. .\"e'!JJ

rort.

Wiiey, 191';'.

(6.1

1::;',anarnse.maT!lTl

li.:.. Introduction

;0

Analytic .Vr;mbf!r Theory, 5pnnger.

~ ·~r~·ag.

: 968.

[7) Dauenport H., Higher Arithmetic, London. Hutchison, 1952.

iB!

Dickson L. E., lntrodution to the Theor] oj .Vumaers. Chicago C'niv. P~ss. 1929.

!9} Est=ann T., lntroduct:on :0 .Vfodern P,..,me .V"moor Theory, Cambr-!dge T.-ac:s '" .ldc.th· ~:nlJ.ttCS . •Vo.

':!. 1952.

!ll i Four.'"ey E., Recreacttons .4.nt.h.m..etz'q!J.e3, T;-o!steme

:12.'

~Ga.-mma·· iO!J.Mc.i. Unsoit.'~d

P:oblem.s Comer,

Edit:on,

3r~o!!.

V~ioer:

:985.

e!

.Vony, Pl.i'.s. : 90~.

52

Goodstein, rt. L., Recursive Recursive Arithmetic

i~

Nu~er

Theory.

A

Develc~8e~~

a Logic-Free Equation

~;

Calc~l~s,

North-Holland Publishing Company, 1964.

Grosswald, Emil and Hagis, Peter, Arithmetic Progressions Consisting Only of Primes, Math. Comput. 33, 1343-:352, 1979.

Guy, Richard K., Unsolved Problems in Number Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1981.

Halberstam, H. and Roth, K. F., Sequences, Oxford D.P., 1966.

Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth Edition, 1984.

Hasse, H., Number Theory, Akademie-Verlag, Berlin, 1977.

Landau, Edmund, Elementary Number Theory, with Exercises by Paul T. Bateman and Eugene E. Kohlbecker, Chelsea, New York, 1958.

Mordell, L. J., Diophantine Equations, Academic Press, London, 1969.

Nagell, T., Introduction to Number Theory, New York, Wiley, 1951.

Niven, I., Irrational Numbers, Carus Math. Monographs, Nc. II, Math. Assoc. of America, 1956.

Ogilvy, C. S., Unsolved Problems for the Amateur, Tomorrow's Math., Oxford Univ. Press, New York, 1962.

I .

O~e,

~u~ber

0.,

Theorv- and

I~s

-

Historv,

McGra~-Hi~l,

NeN

York, 1978.

Re90rt of Institute in the Theory of

~umbers,

Cniv. of

Colorado, Boulder, 1959.

Shanks,

Daniel~

Solved and

U~solved

Proble~s

in

NU2be~

Theory, Spartan, Washington, D. C., 1962.

Sierpinski, W., On Some Unsolved

Proble~s

of

Arith~etics,

Scripta Mathematica, Vol. 25, 1960.

Smarandache, Florentin, A Function in the

Numbe~

Theory *,

in Analele Univ. Timisoara, Vol. XVIII, Fasc. 1, 99.

79-88, 1980; M. R. 83c: 10008.

Smarandache, Florentin, Problemes Avec et Sans ...

Problemes~,

Somipress, Fes, Morocco, 1983; M.R. 84k:

00003.

Ulam, S., A Collection of Mathematical Problems, Interscience, New York, 1960.

Vinogradov, I. M., An Introduction to the Theory of

Nurnbe~s,

Translated by Helen Popova, Pergamon Press, London and New York, 1955.

Florentin Smarandache, Department of Mathematics, N. Balcescu College, Craiova.

[Presented at the 14th American Romanian Academy Annual Convention, held in Los Angeles, California, hosted by the University of Southern California, from April 20 to Aoril 22, 1989.

An abstract was published by Prof. Constantin

54

Cord unea nu, Depa rtmen t of Math emat ics, Univ ersit y of Texa s at Arlin gton , in "Lib ertas Math emat ica," tomu s IX, p. 175, The Grid , Arlin gton , Texa s. Anot her abst ract had been publ ished in the proc eedin gs of the shor t

co~~unications,

Inte rnat iona l Cong ress of Math emat ician s, Berk eley, Cali forn ia, 1986 .)

55

SOLVING PROBLEMS BY USING A FUNCTION IN THE NUMBER THEORY

Let n

~

I, h

~

I, and a

~

2 be integers.

values of a and n is (n + h)! a multiple of an (A generalization of the problem nO

=

For

~hich

?

1270, Mathematics

Magazine, Vol. 60, No.3, June 1987, p. 179, proposed by Roger B. Eggleton, The University of Newcastle, Australia.)

solution (For h = 1 the problem nO = 1270 is obtained.)

ยง1.

Introduction We have constructed a function ry (see [1]) having the

following properties: (a)

For each non-null integer n, ry(n)! is a multiple

(b)

ry(n)

of n; is the smallest natural number with the

property (a).

It is easy to prove: Lemma 1. in the form:

(~)

k, P

~

N*, p .โ ข 1, k is uniquely

writte~

56

k

=

(p)

t, an

(p)

+ ... + t~ an ~

?..;her e

-

1)

/

n 1 > n 2 > ... > n 2,

... ,

e. -

.•. , e., e.

€

1,

(p -

1),

i

=

> 0 and 1 < t

t

1 <

t~

< p,

n i'

1, 2,

j

... , Z,

< p ti

€

N,

1,

j

=

1,

i

=

1,

2,

N*.

We have cons truct ed the func tion ryp' p prim e> 0, ryp

N* -

N*, thus :

=

pn, and

=

Of cour se: Lemma 2. (a)

('i)

(b)

ryp

(a).

k

(k)

€

N *,

ry p

(

k)

=

"'pl: .1.

•

is the smal lest numb er with the prop erty

Now, we cons truct anot her func tion:

ry

Z\{O} - N defin ed as follo ws:

57

=

ry(±. 1)

as

n

(V)

Pi

0,

•

Ps

',.;i th

for i . j, all a i

Pj

= max

( T7 i'

(

E

E

=

±. 1, Pi prime and

N*, T7(n) =

a i) } •

l=£i=£s

It is not difficult to prove T7 has the demanded properties of

§l.

§2.

a, Now, let a = P,

Pi distinct primes.

T7(a)

as

Ps

,with all a i E N* and all

By the previous theory we have:

=

max l=£i=£s

{n p.

1

(a i )}

= T7p

(a)

(by notation).

Hence We

know:

nt

n, p

n, +

(t,P

We

...

put:

n, t, P

+ tt P

nt

)

=

y!p

-1 +

t, p-1

...

+ tt

P

-1

p-1

55

and

.'-,

n,

p

-1 ~

..

nt P

-1 = a n.

'-t

p-1

p-1

Whence

nt

n,

1

-1

p

a

p-1

P

-

+ tt

-1

n, > t, p

+

+ ....'-( p

nt

p-1

.. j..,

or

n, ( 1)

a

(p -

1)

h > (a p -

a -

1)

[t, p

On this condition we take no = t, P

(see Lemma 1), hence n

Consider giving a

~

2, we have a finite number of n.

There are an infinite number of n if and only if a p - a -

=

0, i.e., a

ยง3.

=

1 and p

=

2, i.e., a

Particular Case If h

=

1 and a

~

2, because

=

2.

=

rl ..

~t

and

t.

(1')

(a p

(1")

nt

> p

i. t.

~

I

whic~

p

-

(a p

>

a)

-

is ir.tpossible.

a

nt > 1

follo~"s

-

(

,

\

-}

1 = a

1

1)

If h -

fron

=

1 and a

p -

Ct.,

2 then a

= 1, P = 2,

1 > t. + I

hence

t

=

=

2

=

ti

1,

n1 -

Examole 1.

1,

=

1 whence n

t, P

n1

n, e N* (the solution to proble~ 1270)

Let h

=

= r

16 and a

,

,

. 5-.

Find all n

such that (n +

16)

solution 17

=

17 s

(2)

(2025)

=

17

=

max {17 3 (4), 17 s (2)}

(52).

128 > 7 [t 1 5

n1

Whence a

=

+ ... + tt 5

n, Because 5" > 128 and 7 [ti 5 -t-= 1,

2,

nt

P

]

= =

:=tax {9, IO} = 10 =

5.

Fro~

+ ti .. . .. +

(1) Ne have:

60

n, 128 > 7 t, S

+ t,

whence n: .:::. 1, i. e., n,

=

=

,

1, and t,

=

=

1.

16 < 0, hence we take n

t, 5 -

1, 2,

3.

Then no

=

Exam'Ole 2

But

(n + 7) !

=

" ::1

3 n when n

=

1, 2, 3 , 4, S.

(n + 7) !

=

M Sn when n

l.

(n + 7) !

=

M 7 n when n

= =

(n + 7) !

..

~f

(n + 7) !

=

M 2 n when

no

=

..'-,

n, 2

t, ,

+

•••

.'On ,

n( + t( 2

t(_,

=

1 < t( < 2,

and

n

=

for p prime > 7,

...

I

l.

-

('0')

n

€

N*.

7,

1,

t, +

...

+ t( < 7

no > 0;

In:"

no < O.

etc.

Exercise for Readers If n

€

(n

7)

N*, a

€

N*\(l}, find all values of a and n such

that: +

be a multiple of an.

"

Some Unsolved Problems (see [2]) Solve the diophantine equations: (1)

1']

(x)

â&#x20AC;˘

(2)

1']

(x) = y!

(3)

Conjecture:

1']

(y)

=

1']

(x + y) .

(A sotution: the equation

x = 9, Y = 3).

1']

(x)

=

1']

(x + 1) has

no solution.

References [lJ

Florentin Smarandache, "A Function in the Number Theory," Analele Univ. Timisoara, Fasc. 1, Vol. XVIII, pp. 79-88, 1980, MR: 83c: 10008.

[2J

Idem, Un Infinity of Unsolved Problems Concerning a Function in Number Theory, International Congress of M~thematicians,

Univ. of Berkeley, CA, August 3-11, 1986.

Florentin Smarandache

[A comment about this generalization was published in "Mathematics Magazine", Vol. 61, No.3, June 1988, p. 202: "Srnarandache considered the gen,ral problem of finding positive integers n, a, and k, so that (n + k)! should be a multiple of an.

Also, for positive integers p and k, with p

prime, he found a formula for determining the smallest integer f(k) with the'property that (f(k))! is a multiple of pic.,,]

SOME LINEAR EQUATIONS INVOLVING A FUNCTION IN THE NUMBER THEORY

We have constructed a function ry which associates to eac non-null integer

~

the smallest positive n such that n! is a

mUltiple of m. (a)

= n, where n € N. (mx) = x, where m € Z.

Solve the equation ry (x)

*(b) Solve the equation ry Discussion. (c)

Let rye;) note ry a ry

0

•••

a ry of i times.

Prove tha

there is a k for which

ry('()

(m)

=

ry(,(+l)

(m)

=

n m, for all m

€

Z*\ {I} .

**Find nm and the smallest k with this property.

Solution (a)

The cases n

=

0, 1 are trivial.

We note the increasing sequence of primes less or equal than n by P 1 , Pz , ••• , P,(, and

13 t =

!:

n

[n/pc]

, t

=

I,

2,

... , k;

h~1

where [y] is the greatest integer less or equal than y.

a·I o· ~ 1 S

=

Le': n

whe:r-e all o.1

S

~

a:r-e d':stinc"t

•

Of course we have n < x < n!

=

Thus x t

=

1, 2,

j

€

{1, 2,

a, 0 ~

1

... , k and there exists at least a ... , s} for which

{3.1

-1

(J

••• ,

-

i

I-'

a

+ 1} _

i .

J

Clearly n! is a multiple of x, and is the smallest one. (b)

See [ 1] tao.

Lemma 1m

=

(m)

T']

We consider m

< rn, and

=

(rn )

T']

€

N*.

rn if and only if

4 or rn is a prime.

Of course m! is a multiple of m.

If m

~4

and m is not a prime, the Lemma is equivalent

to there are rn"

m2 such that

and (2 rn 2 < rn or 2 rn, < rn). respectively

T']

Lemma 2.

(m)

< max

TIt

=

rn i

Whence 2~}

{rn 2 ,

(p)

Analogously: 1. < x, .$. x 2 and

(2

< 2 rn 2 < m,

(p x)

=

x

x is not a prime and x .. 2p - x

=

x~

Let p be a prime

T']

(::l.)

T']

< m.

~

and only if x is a prime> p, ,or x Proof:

m2 with 1 < rn, .$. m2

•

= x~

p.

< x~

5.

=

Then

T']

,~

2p.

Hence x > p.

I

x2

•

P, I and 2 x, _< 'x)

ry

(p x)

x z' ~

< max (p,

2 x 2 ) < x respec1:ively 7']

(p x) < max {P, 2

x~,

xzi

< x.

Observations

=

7'] (2 x)

= x - x

(3 x)

7']

x - x

Lemma 3.

=

4 or x is an odd prime.

=

4,6, 9 or

If (m, x)

Of course, 7'] (mx) And x

~

7']

=

1 then x is a prime> 7']

= max

(m), because if x

x is a prime> 3.

(m), 7'] (x)}

{7']

=

(m).

= 7'] (x) = x.

(m) then m • 7'] (m) divides

7']

~

7'] (m)! that is m divides (7'] (m) - 1)! whence 7'] (m) -

1.

Lemma 4. and x

=

2 7']

Proof: ~

7'] (m) -

x2 ' X

If x is not a prime then 7'](m) < x < 2 7'] (m)

(m) if and only if

7']

(m) is a prime.

If x > 2 7'] (m) there are x, ,

= x, x 2 •

multiple of m x.

Xz with

For x, < 7'] (m) we have (x

-

1 < x, <

1) ! is a

Same proof for other cases.

Let x = 2 7'] (m) ; if 7'] (m) is not a prime, then

x = 2 a b, 1 < a < b, but the product + 2)

.•.

(217

(m)

(7']

(m) + 1)

(7'] (m) +

1) is divided by x.

If 7'] (m) is a prime, 7'] (m) divides m, whence

m •

is divided by 7'] (m)2, it results in 7'] (m • 2 7'] (m» '7'](m) ,

but (77 (m) + 1)

(77 (m) + 2)

mUltiple of 2 7'] (m), that is 77 (m • 2

(2 7']

77 (m» (m»

=

2 7'](m)

> 2 •

is a 2

7']

(m).

65

Conclusion All x, prine If ry (0)

nu~ber

> ry

(m), are solutions.

=

is prime,. then x

*If x is not a prime,

ry

(m)

2

ry

(m)

is a solution.

< x < 2 ry

(m), and x does

not divide (x - 1) !jm then x is a solution (semi-open If m

question) .

=

it adds x = 9 too.

J

(No other

solu~icn

exists yet,) ( c)

Lemma 5.

ry

~

(a b)

=

Of course, ry (a) b')! =b'! ,cab)

~

ry

+ ry

(a)

(b).

a' and ry (b) = b' involves (a' â&#x20AC;˘

(b' + 1)

(b'

+ a').

Let a'

~

b'.

Then

a' + b', because the product of a' consecutive

positive integers is a multiple of a'!

Clearly, if m is a prime then k = 1 and nm If m is not a prime then for which

(m) =

ryCk)

ryCk+1J

If m '" 1 then 2 Lemma 6.

If n en

=

~

nm

ry

= m.

(m) < m, whence there is a k

(m). ~

m.

nm = 4 or nm is a prime. n1

< n In .

Absurd.

(**) This question remains, open.

Reference [1]

F. Smarandache, A Function in the Number Theory, An. Univ.

Tim~soara,

seria st. mat., Vol. XVIII, fasc. 1,

pp. 79-88, 1980; Mathematical Reviews: 8Jc: 10008.

Flor entin Smar anda che

[Pub lishe d on "Gamma'.' Jour nal, "Ste agul Rosu " Colle ge, Bras ov, 1987 .J

CONTENTS

R. Muller, Editorial

1

F. Smarandache, A Function in the Number Theory........

3

Idem, An Infinity of Unsolved Problems concerning a Function in the Number Theory.................

12

Idem, Solving Problems by using a Function in the Numbe r Theo ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Idem, Some Linear Equations Involving a Function in the Number Theory... .... .... .................

62

A collection of papers concerninq Smarandache type functions, numbers, sequences, inteqer algorithms, paradoxes, experimental geometries, algebraic structures, neutrosophic probability, set, and logic, etc.