{"id":{"repo_id":"unlv","oai_identifier":"oai:oasis.library.unlv.edu:rtds-2250"},"canonical_url":"https://search.dev.ndltd.org/etd/unlv/oai:oasis.library.unlv.edu:rtds-2250","repository":{"repo_id":"unlv","name":"University of Nevada - Las Vegas","base_url":"https://oasis.library.unlv.edu/do/oai/"},"display":{"title":"Questions and conjectures about multinomial coefficients","abstract":"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 ≤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 ≥ 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 ≡ Rk(m) mod Bk and 0 < Rk( m) ≤ 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 ≤ ai ≤ 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 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 ≥ 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 ≡ Rk(m) mod Bk and 0 < Rk( m) ≤ 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 ≤ ai ≤ 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 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 ≥ 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 ≡ Rk(m) mod Bk and 0 < Rk( m) ≤ 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 ≤ ai ≤ 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