A NOTE ON THE P-ARY REPRESENTATION OF INTEGERS
by
S. Ikeda
and
Dept. of Statistics,
Nihon Univ.
and
I. Maeda
Dept. of Economics,
Dokkyo College
Institute of Statistics Mimeo Series 580
May 1968
This research was partially supported by
U.S. Army Grant No. 325-ARM-I and the Nihon
University.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
A NOTE ON THE P-ARY REPRESENTATION OF INTEGERS
By
S. Ikeda
and
I. Maeda
Dept. of Statistics,
Nihon Univ.
and
Dept. of Economics,
Dokkyo College
This article gives a new form of the p-ary representation of any nonnegative integer for prime p, which is a generalization of Ikeda's formula
for p = 2
[lJ.
For any non-negative integer N, the usual formula of the binary represent at ion is given by
s
L:
N =
k=o
where Ck = [:k] -
2 [2~+1]
,k=o, l,oo.,s; s = max [h;
2h~N}.
On the other hand, Ikeda [lJ showed that N is expanded in the form
where, for any positive integer R, R
2
=
0
if R is even, and = 1 if R is odd.
This implies, by the uniqueness of the binary representation, that
0)
N
2
k
=
(mod. 2).
The purpose of this article is to prove the following identity.
[Nk]
(2)
~+l]
P
- P[
P
= (N )
P
k P
or, equivalently,
[:k]
(3)
= (N )
P
(mod. p)
k
,
where Nand k are non-negative integers, p is any given prime, [ ] is the
ordinary Gauss symbol, and (N)
denotes the remainder when (N) is divided
n p
by p.
of N,
n
Here, we use the convention that the number of n-combinations out
is equal to zero i f n > N •
PROOF OF (2)
(1)
k
In the case 0 ~ N <p , it is evident that
= 0 = (N )
P
k P
for any non-negative integer k.
(II)
In the case pk~
N<
pk+l, N is expressed uniquely in the following
form
N =
1Sj
for some integers j and i;
-
jpk + i,
~ p-l and 0 ~ is pk_1.
-
-
Then, for the left hand side of (2) it is easy to see that
[
for any non-negative k.
ok +k oJ
JP
J.
.
_
[ok
JP + oJ
k+l
P
p
Thus, putting
1
k
0
°
= (N
J,J.
P
= (JP k
°
)
k P
I
k
°
j , J.
+
p
we have to show that the relation
( 4)
= JO
J.
P
=j
k
°
J.
holds true for any non-negative integer k.
In the first place, we shall prove the relation (4) when i=o, that is,
I
. k
k
= j, or equivalently, ( JP
k ) = ~"
j,o
(mod. p).
P
Since
.k
. k
(JP )
k
.(JP
J k
=
P
P
1
-
)
- 1
'
it suffices to prove that the relation
. k
(5)
1
(JP
-
P
- 1
k
=
1
-
)
holds true for any non-negative k.
(mod. p)
When k =
0,
this relation is trivial.
Let us consider the expansion
. k
(JP
(6)
P
k
-
1
k
P -1. k
) =
IT
- 1
p---.,:---h
h
...;..;J
h=l
for any positive k.
In the above expansion, h is expressed in general as
where 6 (h) is prime to p and m(h) is an integer such that 0 S-m(h) ~ k-l.
-
-
If h is prime to p, then m(h) = 0 and hence h = 6(h).
Thus, it follows from (6) that
k
P -1
11
(jpk-m(h) - 6 (h) )
h=l
(7)
k
P -1
7T
6(h)
h=l
Here, we have
k
P -1
7T
h=l
k
(jpk-m(h) _ 6(h) ) = p.Q(p) + (-l)P -1
where Q(p) is a polynomial in p.
k
P -1
17
h=l
6(h) ,
Thus, we obtain
( jpk
k
P
_ 1
- 1
k
p.Q(p)
) = -:-=:..-..;~k
+ (-l)P -1
P -1
Tf
6(h)
h=l
k
p~3,
Recalling that p -1 is even when
. k
1
(JP - )
k
P - 1
we thus have,
p.Q(p)
= ~:..-..;;.;..:..:;_
k
+ L
P -1
IT
6(h)
h=l
k
P -1
Since
IT
6(h) is prime to p, it divides Q(p), and hence we have (5)
h=l
for p ~ 3.
When p = 2, it holds that
- 1
~
1
(mod. 2).
Thus, the relation (5) is true for any prime p.
In the second place, we shall prove the relations
k
(8)
k
I j , 1 = I j , i-I
for i=1,2, .•. ,pk_l and j=1,2, ..• ,p-l, (k
~l).
Putting, as before, h = pm(h) .6(h) for h=1,2, .•• ,pk_ l , we have
k
. k + i ) _ (JP
. k + 1..
(JP
k
k
p
P
-
1) =
k
=p
k-m(i)
J'pk + i - h
h
h=l
P -1
j
P7T- l
IT
. k-m(h) +
( JP
h=l,h~i
k
P -1
1T
h=l
p
m(i)-m(h)
6(i) - 5(h) )
k
P -1
if 6(h) is prime to p and OSm(i) < k-l, the second factor of the
=
h=l
last expression of the above equalities should be a positive integer.
Since
Hence
(jpk +k i - 1) = 0
(mod. p),
p
which proves (8).
This completes the proof of (4) in the case (II).
(III)
In the case pk+l~ N, we may put for some q, j and i,
N = qp
k+l
. k
+ JP + i,
where q is a positive integer, j=o,l, •.• ,p-l and i=o,l, •.• ,pk_l •
Let us put
Jk
= (qp
J,i
k+l + jpk
+ i)
k
P
P
Then, by a similar argument in the case (II) , we have
k
k
=
= Jj,l =
J.
k l'
J,P -
j=o, 1, ••. ,p-l
for any k positive and
Jk
j,o
for any non-negative k.
= j,
j=O, 1, .•. ,p-l
This implies that (2) is true in the case (III).
Hence, in all the cases, we proved the relations (2) for any prime
number p,and non-negative integer k.
REFFERENCE
[lJ
S. Ikeka,
"A note on the Hamming weight of binary vectors", to appear
in the Journal of the College of Industrial Technology,
Nihon University, Tokyo, Vol.2, No.1 (1968).