Questions and conjectures about multinomial coefficients

The purpose of this thesis is to try to answer some of the questions in Dr. Bachman's paper "On Divisibility Properties of Certain Multinomial Coefficients". First we let {ai} be any sequence (finite or infinite) of positive integers such that i1ai &le;1 . It is clear that n!&sqbl0;na1 &sqbr0;!&sqbl0;na2&sqbr0; !&sqbl0;na3&sqbr0;!&ldots; is an integer because it is a multiple of a certain multinomial coefficient. We let fan=n! Ln&sqbl0;n a1&sqbr0;!&sqbl0;na 2&sqbr0;!&sqbl0;na3 &sqbr0;!&ldots; where L(n) = lcm(1, 2, 3, .., n). It is easy to show that fa(n) is integer-valued. In particular, we would like to study the sequence a1 = b1 = 2 and ak+1 = bk+1 = Pki=1 bi + 1. The first goal of my thesis was to prove the following conjecture by computer for all m up to 100; Conjecture 1. For every positive integer m there exists a number n0 such that m divides f( n) for all n > n0 where fn=n!L n&sqbl0;n2&sqbr0; !&sqbl0;n3&sqbr0;!&sqbl0;n 7&sqbr0;!&ldots I did this by using Theorem 1 of Dr. Bachman's paper; Theorem 1. pv|| f(n) if and only if there are exactly v pairs of integers (k,l),k,l &ge; 1, such that Rk&parl0;&sqbl0;npl &sqbr0;&parr0;Bk< Rk+1&parl0;&sqbl0;npl &sqbr0;&parr0;Bk+1 with Rk(m) defined as m &equiv; Rk(m) mod Bk and 0 < Rk( m) &le; Bk where Bk = bk+1 - 1; The second part of my thesis is concerned with attacking Conjecture 1 as it was written in Dr. Bachman's paper. Before we can restate Conjecture 1 we need to define the base p expansion of a positive integer. We write nj = a0pj + a1pj -1 +..+ aj where 0 &le; ai &le; p - 1. Now we restate Conjecture 1 as Conjecture 2; Conjecture 2. Let {nj} be defined above. Then there exist infinitely many integers j for which the inequality Rk&parl0;nj&parr0;B k<Rk+1&parl0;nj &parr0;Bk+1 holds for some integer k = k( j); In my thesis, I will give proofs of Conjecture 1 for special { nj}. I discovered these proofs together with Dr. Bachman and Theorem 1 will be used in all the proofs.

Name thesis:degree_name

Master of Science (MS)

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

University of Nevada, Las Vegas

Year

2001

Author dc:creator

• Kessler, Troy Richard

Contributors dc:contributor

• Gennady Bachman

Statement dc:rights

• IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

Language dc:language

English

Identifier

https://oasis.library.unlv.edu/rtds/1251

OAI identifier oai:identifier

oai:oasis.library.unlv.edu:rtds-2250