Combinatorial approaches to integer sequences

Malouf, Janice L.

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https://hdl.handle.net/2142/21236 Copy

Description

Title

Combinatorial approaches to integer sequences

Author(s)

Malouf, Janice L.

Issue Date

1994

Doctoral Committee Chair(s)

Halberstam, Heini

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

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.

Type of Resource

text

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http://hdl.handle.net/2142/21236

Copyright and License Information

Copyright 1994 Malouf, Janice L.

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