Two theorems on the addition of residue classes
It is well known that if a 1,…, a m are residues modulo n and m ⩾ n then some sum a i1 + ⋯ + a ik , i 1 < ⋯ < i k , is 0 (mod n). In recent related work, Sydney Bulman-Fleming and Edward T.H. Wang have studied what they call n-divisible subsequences of a finite sequence σ, and made a number of...
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| Published in | Discrete mathematics Vol. 81; no. 1; pp. 11 - 18 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.04.1990
|
| Online Access | Get full text |
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| Summary: | It is well known that if
a
1,…,
a
m
are residues modulo
n and
m ⩾
n then some sum
a
i1
+ ⋯ +
a
ik
,
i
1 < ⋯ <
i
k
, is 0 (mod
n). In recent related work, Sydney Bulman-Fleming and Edward T.H. Wang have studied what they call
n-divisible subsequences of a finite sequence σ, and made a number of conjectures. We confirm two of those conjectures in a more general form. Let
f(
a
1,…,
a
m
;
j be the number of sums formed from the
a
i
which are congruent to
j (mod
n). We prove two main theorems: 1.
If f(
a
1,…,
a
m
; 0) < 2
m−1
then f (
a
1,hellip;,
a
m
; 0) ⩽ 3 · 2
m−3
; 2.
Let m ⩾
n.
There exist a
1,hellip,
a
m
for which f(
a
1,hellip:,
a
m
;
j)
is odd if and only if n itis not a power of 2. |
|---|---|
| ISSN: | 0012-365X 1872-681X |
| DOI: | 10.1016/0012-365X(90)90174-G |