自然科學與教育
2015,第一期第二卷,63-64
Natural Science and Education
2015,1(2),63-64
A Generalized Basis Representation Theorem
Kuo-Jye Chen
Department of Mathematics, National Changhua University of Education
[email protected]
We present a generalized basis representation theorem.
Key words: basis representation
Classiciation :Mathematics
In this note, we will present a generalized basis representation theorem (Theorem 1 below).
The proof uses Andrews’ s idea [1] and the following simple lemmas.
Lemma 1. Let a0 , a1 , a2 , … be a sequence of positive integers. Then
(1  a0 )(1  a1 )(1  a2 )  (1  an )
 1  a0  (1  a0 )a1  (1  a0 )(1  a1 )a2    (1  a0 )(1  a1 )(1  a2 )  (1  an1 )an .
Lemma 2. Let a0 , a1 , a2 , … be a sequence of positive integers. Then, for any integer n  0,
n  1  a0 1  a1 1  a2   1  a j 1  .
These two lemmas can be readily proved by induction. Our main theorem is
Theorem 1. (Generalized Basis Representation Theorem)
Let a0 , a1 , a2 , … be a sequence of positive integers. Let B j be defined by B0  1 and B j =
1  a0 1  a1 1  a2   1  a j 1  for
j  1. Then every integer n  0 can be uniquely
expressed in the form
Where d j
n  d0 B0  d1B1  d2 B2  ,
is an integer satisfying 0  d j  a j .
Proof. Let R  n  denote the number of representations of a nonnegative integer n in
the form (1). Note that R  0   1 . Let n  1 and assume that
n  d0 B0  d1B1  d2 B2    0  d j  a j 
is a representation for n .
If d0  0, then
n  1   d0  1 B0  d1B1  d2 B2   .  0  d j  a j , d0  1
is a representation for n  1.
If d0  0, then there exists a positive integer m such that
n  dm Bm  dm 1Bm1  dm 2 Bm 2   .
0  d
j
 a j , m  1, dm  1
It follows from Lemma 1 that
n  1   Bm  1   d m  1 Bm  d m 1Bm1  
 a0 B0  a1 B1  a2 B2    am1Bm1   d m  1 Bm  d m1Bm1   ,
which is a representation for n  1.
(1)
A Generalized Basis Representation Theorem
Thus we have shown that
R  n   R  n  1 .  n  1
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 2
Note that (2) holds even if n has no representation.
It follows from (2) and Lemma 2 that
1  R  Bn   R  n   R  n  1  R  0   1.
Therefore
R  n   1,  n  0 
which completes the proof of the theorem.□
Specializations.
Two well-known basis representation theorems follow as special cases.
Corollary 1. Let k be a positive integer  2 . Then every integer n  0 can be uniquely
written as
n  d0  d1k  d2 k 2  d3k 3  ,
where d j is an integer satisfying 0  d j  k  1.
Proof. In Theorem 1, put a j  k  1 , for all j  0. □
Corollary 2. Every integer n  0 can be uniquely written as
n  d01! d1 2! d2 3! d3 4!  ,
where d j is an integer satisfying 0  d j  j  1.
Proof. In Theorem 1, put a j  j  1 , for all j  0. □
References
1.Andrews, George E., On radix representation and the Euclidean algorithm, Amer. Math. Monthly 76
(1969), pages 66-68.