# The Postage Stamp Problem and Its Applications

## Date

2014-05

## Authors

Song, Zhaochen

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## Abstract

Let A = { 0 = a0 < a1 < a2 < · · · < ak} be a set of k + l nonnegative integers. Let n(h, A) denote the largest positive integer n so that every integer x E [O, n] can be written as a sum of exactly h not necessarily distinct elements from A, where [a, b] denotes the set of integers x : a ~ :r; :s; b. Let k be a positive integers. Define
n(h, k) = max{n(h, A) I AC N and IAI = k + 1},
where N is the set of all nonnegative integers. Using the definition of n(h, A), we can further rewrite n(h, k) as
n(h, k) = max{n I AC N, IAi = k + 1, and hA ~ [O, n]}.
The postage stamp problem is the study of n(h, k) and other related problems. In this thesis, we mainly focus on postage stamp problem and some extremal bases for finite cyclic groups, a closely related problem with applications in information network design.
We will introduce some extremal functions, namely n(h, 2) and n(h, 3), and discuss some simple properties and estimates of them. This will further expand to the discussion of some other bases and extremal functions with lager k or n values. Some other closely related problem, such as the extremal bases for finite cyclic groups and the Frobenius coin problem, will also be studied in this thesis.

## Description

## Keywords

number theory, additive functions, finite groups

## Citation

Song, Z. (2014). The postage stamp problem and its applications (Unpublished thesis). Texas State University, San Marcos, Texas.