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Title:Combinatorial approaches to integer sequences
Author(s):Malouf, Janice L.
Doctoral Committee Chair(s):Halberstam, Heini
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Combinatorial methods are used to prove several results in number theory. The chapters may be read independently, and are briefly discussed below.
In 1935 Erdos proved that every additive basis $\rm{\cal B}$ of order h is an essential component by establishing the inequality $\rm \sigma({\cal A} + {\cal B})\ge \sigma({\cal A}) + {1\over {2h}}\sigma({\cal A})(1-\sigma({\cal A})),$ where $\rm\sigma({\cal A})$ denotes the Schnirelmann density of $\rm{\cal A}$. This lower bound was improved by Helmut Plunnecke in 1970 to $\sigma({\cal A} + {\cal B})\ge\sigma ({\cal A})\sp{1-1/h}$ using an application of graph theory. A simplification of Plunnecke's proof is presented in Chapter 1.
The sequence of numbers $\{ a\sb{i}\}$ defined by the recurrence $a\sb{n} = (a\sb{n-3}a\sb{n-1} + a\sbsp{n-2}{2})/a\sb{n-4}$ for n $>$ 3, with initial values $a\sb0, a\sb1, a\sb2, a\sb3$ = 1, is shown to be integral in Chapter 2. The proof is extended to address more general sequences of this type.
In a famous work so entitled, Erdos and Selfridge established that the product of consecutive integers is never a power. In Chapter 3 related problems are considered in which one starts with n, not a kth power and selects a set of integers larger than n whose product with n forms a kth power, seeking to minimize the largest number used. In the restricted problem, the condition is placed on gaps between integers chosen so that no k consecutive numbers are omitted. In the case of squares, it is shown that the largest number used will not exceed 3n $-$ 3.
A set of integers is called sum-free if it contains no solution to the equation x + y = z. Erdos showed that every set of n integers has a sum-free subset with at least n/3 elements. This was strengthened by Alon and Kleitman to $>$n/3, and they showed by means of an example that the 1/3 cannot be improved to any number as large as 12/29(=.4137$\...$). A construction is given in Chapter 4 which shows that it cannot be improved to 2/5.
A well-known theorem of Pillai and Szekeres states that for k $\le$ 16, every set of k consecutive integers contains one which is relatively prime to the others. It was established by Brauer and Pillai that this is false for k $>$ 17. The condition of coprimality is strengthened to require that each of the numbers $\{n,n + 1,\..., n + k\}$ have a factor in common with either n or n + k, and values of k for which this is possible are studied in Chapter 5.
Issue Date:1994
Rights Information:Copyright 1994 Malouf, Janice L.
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9512476
OCLC Identifier:(UMI)AAI9512476

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