The postage stamp problem and arithmetic in base
Czechoslovak Mathematical Journal (2008)
- Volume: 58, Issue: 4, page 1097-1100
- ISSN: 0011-4642
Access Full Article
topAbstract
topHow to cite
topTripathi, Amitabha. "The postage stamp problem and arithmetic in base $r$." Czechoslovak Mathematical Journal 58.4 (2008): 1097-1100. <http://eudml.org/doc/37888>.
@article{Tripathi2008,
abstract = {Let $h,k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $hA:=\lbrace a_1+a_2+\cdots +a_r\colon a_i \in A, r \le h\rbrace $ 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 $\lbrace 1,2,\ldots ,n\rbrace \subseteq hA$. 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 $k=3$.},
author = {Tripathi, Amitabha},
journal = {Czechoslovak Mathematical Journal},
keywords = {$h$-basis; extremal $h$-basis; geometric progression; -basis; extremal -basis; geometric progression},
language = {eng},
number = {4},
pages = {1097-1100},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The postage stamp problem and arithmetic in base $r$},
url = {http://eudml.org/doc/37888},
volume = {58},
year = {2008},
}
TY - JOUR
AU - Tripathi, Amitabha
TI - The postage stamp problem and arithmetic in base $r$
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1097
EP - 1100
AB - Let $h,k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $hA:=\lbrace a_1+a_2+\cdots +a_r\colon a_i \in A, r \le h\rbrace $ 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 $\lbrace 1,2,\ldots ,n\rbrace \subseteq hA$. 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 $k=3$.
LA - eng
KW - $h$-basis; extremal $h$-basis; geometric progression; -basis; extremal -basis; geometric progression
UR - http://eudml.org/doc/37888
ER -
References
top- Alter, R., Barnett, J. A., 10.2307/2321610, Amer. Math. Monthly 87 206-210 (1980). (1980) Zbl0432.10032MR1539314DOI10.2307/2321610
- Hofmeister, G., Asymptotische Abschätzungen für dreielementige Extremalbasen in natürlichen Zahlen, J. reine angew. Math. 232 77-101 (1968). (1968) Zbl0165.06201MR0232745
- Rohrbach, H., 10.1007/BF01160061, Math. Z. 42 1-30 (1937). (1937) MR1545658DOI10.1007/BF01160061
- 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). (1974) MR0371669
- Stöhr, A., Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, I, J. reine Angew. Math. 194 40-65 (1955). (1955) MR0075228
- Stöhr, A., Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, II, J. reine Angew. Math. 194 111-140 (1955). (1955) MR0075228
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.