The postage stamp problem and arithmetic in base r

Amitabha Tripathi

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1097-1100
  • ISSN: 0011-4642

Abstract

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Let h , k be fixed positive integers, and let A be any set of positive integers. Let h A : = { a 1 + a 2 + + a r : a i A , r 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 , ... , n } h A . Let n ( h , k ) : = max A : 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 .

How to cite

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Tripathi, 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

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  1. Alter, R., Barnett, J. A., 10.2307/2321610, Amer. Math. Monthly 87 206-210 (1980). (1980) Zbl0432.10032MR1539314DOI10.2307/2321610
  2. Hofmeister, G., Asymptotische Abschätzungen für dreielementige Extremalbasen in natürlichen Zahlen, J. reine angew. Math. 232 77-101 (1968). (1968) Zbl0165.06201MR0232745
  3. Rohrbach, H., 10.1007/BF01160061, Math. Z. 42 1-30 (1937). (1937) MR1545658DOI10.1007/BF01160061
  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). (1974) MR0371669
  5. Stöhr, A., Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, I, J. reine Angew. Math. 194 40-65 (1955). (1955) MR0075228
  6. Stöhr, A., Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, II, J. reine Angew. Math. 194 111-140 (1955). (1955) MR0075228

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