1945]
A
SET
OF EIGHT
NUMBERS
379
it can
be shown
for large
n that Y,_1 is approximately
equal to (2irnpq)-112,
or
1//V2irau.
Thus
from (6)
we obtain
the approximation
(I1) MD
n
=
X/2/7ran
=
0.79788a,,.
More
exact computation,
using the
remainder
terms
in Stirling's
formula,
yields
the better
approximation
(12)
-
(MD,)2
=
npq
+ (np
-
[np])(nq
-
[nq])
-(1-
pq)/6
+
En/24n,
2
where
the error
coefficient
En
becomes numerically
less
than or
equal
to
unity
as n
becomes
infinite,
for
all
choices
of
np
between
1
and
n-1;
and
[np]
and
[nq]
denote the
greatest
integers
not
exceeding
np
and
nq
respectively.
A SET
OF
EIGHT
NUMBERS
ARTHUR
PORGES,
Western
Military
Academy
1.
Introduction.
In
this
paper
the
operation
of
adding
the
squared
digits
of
any
natural
number
A
a finite number
of times
is
proved
to
transform
A
either
to unity
or
to one
of a
set
of eight
natural
numbers
closed
under
the
operation.
2.
Definitions.
We
use
the
expression
natural
number
to denote
a
member
of
the
set 1, 2,
3,
of positive
integers.
Zero
has
not
been
adjoined
to this
set
and
is
not to be included
in
the
definition.
The
operator G
is
defined
by the
equation
R
2
(1)
G(A)
=
X,
i=l
where
A is
a
natural
number
of
R
digits
given
by
R
(2)
A
=
Xloi-1.
Since
A has
R
digits,
XR
0
O.
We
note
that
G(0)
=
0,
and
G(1)
=
1.
Using
the
customary
notation,
we write
G
(A),
where
n> 1,
for
n
successive
applications
of the
operator
G to
A.
G'
is
not a
linear
operator
since,
in
general,
G(A1+A2)OG(A1)+G(A2).
The set
of
numbers
a,=
4,
as=
89,
a2
=
16,
a6=
145,
a3
=
37, a7=
42,
a4=
58,
a8=
20,
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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 number
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 inequalities
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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19451
A
SET OF
EIGHT
NUMBERS
381
THEOREM
1.
For
every
natural
number A
there
exists either
a
positive
integer
n
such
that
(5)
holds
for
all r 2
n,
or a
positive
integer m
such
that
(13)
Gr(A)
=
a'
for
all r
>
m,
where
a' is
some
element
of
Set
K.
From
Lemma 2
it
is
evident we
need
prove
the
theorem
only
for
A
!
162.
The
writer
was
unable
to
find a
simple
indirect
proof
sufficiently
superior
to
the
following
direct
one of
selective
verification
to
justify its
inclusion
here.
We
consider
two
cases.
Case 1.
I00<A
?162.
For
A
thus
restricted, it
is
apparent
that
G(A)
!G(159)
=
107.
Direct
appli-
cation
of
the
operator
G to
A
over
the
range
100
to
107
gives
G(100)
=
1,
G6(104)
=
a'
=
89,
(14)
G2(101)
=
a' =
4,
G3(105)
=
a'
=
16,
G6(102)
=
a' =
89,
G(106)
=
a'
=
37,
G2(103)
=
1,
G6(107)
=
a' =
89,
thus
completing
the
proof
of the
theorem
for
Case
1.
Case 2.
0<A<100.
For
A
=1OX+
Y,
where
0?X:9,
and
0:
Yg9,
the
following
identity
is
valid.
(15)
G(IOX
+
Y)
G(1OY
+ X).
Further,
if
Gn(A)=a',
and
Gm(B)=A,
it
follows
that
there
exists
a
number
h=n+m
such
that
Gh(B)
=
a'.
By
means
of
these
considerations, it
is
possible to
verify
Theorem
1
numeri-
cally
for
all
A
<100 by
actual
computation
of
Gn(A) for
30
values
of
A
<100,
thus
completing
the
proof
of
the
theorem.
The
writer is
aware
of
the
inelegance of
such a
proof,
and
would
like
very
much to
see a
simple
indirect
one.
However,
proving
the
non-existence
of
an-
other
set
like
(3),
which
seems
a
necessary
step,
is
quite
difficult because
of
the
non-linear
character of
G.
COROLLARY.
For
every
natural
number A
there
exists either
a
positive
integer
n
such
that
Gn(A)
=
1,
or a
positive
integer m
such
that
Gm(A)
=4.
The
corollary
follows
directly
from
Theorem
1
and
the
nature
of
Set
K.
Since
every
natural
number
is
transformed
either
into
unity
or
into
an
element
of
Set
K
by the
operator G,
we
need
only
note
that
for
every
a'
$4,
there
exists
a
positive
integer r
<
7
such
that
Gr(a')
=4.
THEOREM
2.
The
number of
digits N
in
G(A),
where A has
R
digits,
satisfies
the
inequality
(16)
N
< 2.9
+
logio
R.
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382
A
SET
OF.
EIGHT
NUMBERS
This
theorem
is a
simple
consequence
of
the
inequality
G(A)
g
81R.
We
have
(iT)
G(A)
?
101'9
+
logio
R
a
number
of
N
digits,
where
N
S
2.9
+
Log1o
R.
THEOREM
3.
The
only
solutions
in
natural
numbers
of
Gn(A)
=
A,
where
n
>
1,
are
(19)
A
=
1,
n
=
J,
(20)
A =
a',
n=8,
where
J
is any
natural
number.
If
we
assume the
existence
of
a
natural
number
A
>
1
and
different
from
a'
such
that
Gn(A)
=A for
some
n
?
1,
it
follows
that
A
would
not
be
transformed
into
either
unity
or an
element
of
Set
K by
a
finite
number
of
applications
of
the
operator
G
to
A. But
this
is a
direct
contradiction
of
Theorem
1,
and hence
the
assumption
is false.
5.
Concluding
Remarks.
A
problem
suggested
by
the one
just discussed
is
that
of
repeatedly
summing
the
cubed
digits
of a natural
number.
A complication
occurs,
however,
since
there
is more
than
one
number
A such
that
H(A)
=A,
where
H
is the
operator
analogous
to
(1)
given
by
R
(21)
H(A)=
Xi
X.
i=1
For example,
H(153)
=
153,
H(407)
=
407,
and
H(371)
=
371.
This
destroys
the
factor
of
uniqueness,
since
H(A)
may
be
unity
as
when
A
=
100;
or A
may
be
transformed
into a number
A'
like 153.
It
is
interesting
to
note
that
since
for
any
number
A
transformed
into
some
element
of
Set
K by
a
finite
number
of
applications
of G
we can
construct
a
number
B
=
-OA
such
that
G(B)
=1,
there
are at
least
"as
many"
numbers
satis-
fying
(5)
as
(13).
This intuitionally
unsatisfying
conclusion
results
from
the
com-
parison
of two infinite
sets.
Leibniz
discovers
the
obvious.
I have
made
some
observations
on
prime
numbers which,
in
mxn
opinion,
are
of
consequence
for the
perfection
of
the science
of numbers
....
If
the
sequence
[of
primes]
were
well
known,
it would
enable
us
to
uncover
the
mystery
of numbers
in
general;
but
up
till
now
it has
seemed
so bizarre
that
nobody
has
succeeded
in finiding
any
affirmative
char-
acteristic
or
property
.
.
...
I
believe
I
have
found
the
right
road
for
penetrating
their
[primes']
nature:
but not
having
had
the
leisure
to
pursue
it,
I
shall
give
you
here
a
positive
property,
which
seems
to
me curious
and
useful.-Leibniz,
in
a
letter to
the editor
of the Journal
des
Savans,
1678.
The discovery:
a
prime
is
necessarily
of one
or other
of the forms
6n+1,
6n
+5.-Contributed.
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