380

A SET

OF EIGHT

NUMBERS

[Aug.-Sept.,

is closed under the operation

defined by (1). We call (3) Set K, and

use the

sym-

bol a' to denote

any non-specified

element

of the set. The

equation

(4)

G8(a')

=

a'

is easily

verified.

Numbers of

the form lOn,

13. 10,

lOn+1+3,

where n is

a positive

integer

or

zero,

and

others not specified

here, satisfy

the equation

(5)

Gt(A)

=

1

for

some integer

r>0.

Any natural

number

satisfying

(5)

will be denoted

by

the

symbol

b'.

3.

Preliminary

Lemmas.

In

what follows,

the

symbols

A and B

always repre-

sent

finite natural

numbers

in the

denary

system

of notation.

LEMMA

1. A

ny natural

number

A of R

digits, where

R 2 4,

satisfies

the inequality

(6)

G(A)

<

A.

It

is evident

that G(A)

g

81R,

and that

A 2 OR-1.

The

inequality

(7)

81R

<

1OR-1

becomes,

upon

taking the

common

logarithm

of

each member

and

transposing,

(8)

logio

R

<R

-

2.9085,

an

inequality

valid

for

R

>4.

LEMMA

2. For any

natural number

A

there exists

a positive

integer

n such that

(9)

Gn(A)

<

162.

For R24,

Lemma

1

establishes

the

inequality

(6).

As a

direct

consequence

of

(6),

the operator

G applied

to

A

a finite numbe r

of

times

must

result

in

a nat-

ural number

of

less

than four

digits,

since

for

R=4, G(A)

9324.

For R <4, the following inequ alities

are

readily

established.

(10)

G(A)

!

243,

(11)

G2(A)

5

G(199)

=

163,

(12)

G3(A)

5

G(99)

=

162.

S?nce

G(A),

where

A is

a three

digit

number,

cannot exceed

3.81 =

243, (10)

is

obviously

valid.

Also,

since

G(199)

2G(B)

for

any

Bg

243, (11)

holds.

Finally,

since

G(99)2G(P)

for

any

Pg163, (12)

is

proved.

The

inequalities

(10),

(11),

and

(12)

complete

the

proof

of

Lemma

2.

4.

Convergence

of

Gn(A).

The

following

theorem

is

the

main result

of

this

paper.

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