| Title:
|
The postage stamp problem and arithmetic in base $r$ (English) |
| Author:
|
Tripathi, Amitabha |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
1097-1100 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $h,k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $hA:=\{a_1+a_2+\cdots +a_r\colon a_i \in A, r \le h\}$ denote the set of all integers representable as a sum of no more than $h$ elements of $A$, and let $n(h,A)$ denote the largest integer $n$ such that $\{1,2,\ldots ,n\} \subseteq hA$. Let $n(h,k):=\max _A\colon n(h,A)$, where the maximum is taken over all sets $A$ with $k$ elements. We determine $n(h,A)$ when the elements of $A$ are in geometric progression. In particular, this results in the evaluation of $n(h,2)$ and yields surprisingly sharp lower bounds for $n(h,k)$, particularly for $k=3$. (English) |
| Keyword:
|
$h$-basis |
| Keyword:
|
extremal $h$-basis |
| Keyword:
|
geometric progression |
| MSC:
|
11B13 |
| MSC:
|
11D04 |
| idZBL:
|
Zbl 1174.11013 |
| idMR:
|
MR2471168 |
| . |
| Date available:
|
2010-07-21T08:11:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140442 |
| . |
| Reference:
|
[1] Alter, R., Barnett, J. A.: A postage stamp problem.Amer. Math. Monthly 87 206-210 (1980). Zbl 0432.10032, MR 1539314, 10.2307/2321610 |
| Reference:
|
[2] Hofmeister, G.: Asymptotische Abschätzungen für dreielementige Extremalbasen in natürlichen Zahlen.J. reine angew. Math. 232 77-101 (1968). Zbl 0165.06201, MR 0232745 |
| Reference:
|
[3] Rohrbach, H.: Ein Beitrag zur additiven Zahlentheorie.Math. Z. 42 1-30 (1937). MR 1545658, 10.1007/BF01160061 |
| Reference:
|
[4] Stanton, R. G., Bate, J. A., Mullin, R. C.: Some tables for the postage stamp problem.Congr. Numer., Proceedings of the Fourth Manitoba Conference on Numerical Mathematics, Winnipeg 12 351-356 (1974). MR 0371669 |
| Reference:
|
[5] Stöhr, A.: Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, I.J. reine Angew. Math. 194 40-65 (1955). MR 0075228 |
| Reference:
|
[6] Stöhr, A.: Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, II.J. reine Angew. Math. 194 111-140 (1955). MR 0075228 |
| . |
