Physical Sciences and Mathematics Commons^™

Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov

Publications and Research

The Erdős–Straus conjecture, initially proposed in 1948 by Paul Erdős and Ernst G. Straus, asks whether the equation 4/n = 1/x + 1/y + 1/z is solvable for all n ∈ N and some x, y, z ∈ N. This problem touches on properties of Egyptian fractions, which had been used in ancient Egyptian mathematics. There exist many partial solutions, mainly in the form of arithmetic progressions and therefore residue classes. In this work we explore partial solutions and aim to expand them.

Fostering Student Discovery And Conjecture In Multivariable Calculus, Aaron Wangberg

Journal of Humanistic Mathematics

Who owns the mathematical ideas in the undergraduate classroom? A traditional mathematics classroom and curriculum imposes several barriers that prevent students from discovering and engaging with mathematical concepts. Definitions, notations, and theorems require mastery before students can work meaningfully with the underlying mathematical concepts. Raising Calculus to the Surface utilizes a different approach by providing students multiple entry points to engage meaningfully with mathematics ideas and allows students to promote meaningful ideas and conjectures into the classroom discourse to formalize their explorations. In this paper, we describe several characteristics built into the project materials, including a rubric designed to encourage …

Combinatorial Polynomial Hirsch Conjecture, Sam Miller

HMC Senior Theses

The Hirsch Conjecture states that for a d-dimensional polytope with n facets, the diameter of the graph of the polytope is at most n-d. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in n and d on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the …

Locally Convex Words And Permutations, Christopher Coscia, Jonathan Dewitt

Dartmouth Scholarship

We introduce some new classes of words and permutations characterized by the second difference condition pi(i - 1) + pi(i + 1) - 2 pi(i)

Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr

Honors College Theses

The Collatz Conjecture is a deceptively difficult problem recently developed in mathematics. In full, the conjecture states: Begin with any positive integer and generate a sequence as follows: If a number is even, divide it by two. Else, multiply by three and add one. Repetition of this process will eventually reach the value 1. Proof or disproof of this seemingly simple conjecture have remained elusive. However, it is known that if the generated Collatz Sequence reaches a cycle other than 4, 2, 1, the conjecture is disproven. This fact has motivated our search for occurrences of 4, 2, 1, and …

Poincare Conjecture Disproven, Proofs And Showings Of No Need To Twist And Bend Manifolds Into Spheres!, James T. Struck

James T Struck

1 Poincare Conjecture Disproven, Proofs and Showings of No Need to Twist and Bend Manifolds Into Spheres! Are 3 dimensional surfaces one to one or homeomorphic with 3 dimensional spheres? Disproofs and lack of correlation, lack of need proofs follow. 1. Take a book and open and close the book. When we close the book, the point on a page does not map directly to the sphere; the point in the closed book travels differently to the sphere not the same travel as when the book is open. 2. We do not need to map the point on the book …

Prove It!, Kenny W. Moran

Journal of Humanistic Mathematics

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.

The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz

Faculty Publications & Research

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S,T) of the vertex set V(G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t . Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.

The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz

Noah Prince

Erdös and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S,T) of the vertex set V(G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t . Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.

Totally Real Galois Representations In Characteristic 2 And Arithmetic Cohomology, Heather Aurora Florence De Melo

Theses and Dissertations

The purpose of this paper is to provide new examples supporting a conjecture of Ash, Doud, and Pollack. This conjecture involves Galois representations taking Gal(Q bar/Q) to the general linear group of 3 x 3 matrices in characterisic 2, and our examples are where complex conjugation is mapped to the identity. Since this case has not yet been examined, the results of this paper are quite significant.

Explicit Computations Supporting A Generalization Of Serre's Conjecture, Brian Francis Hansen

Theses and Dissertations

Serre's conjecture on the modularity of Galois representations makes a connection between two-dimensional Galois representations and modular forms. A conjecture by Ash, Doud, and Pollack generalizes Serre's to higher-dimensional Galois representations. In this paper we discuss an explicit computational example supporting the generalized claim. An ambiguity in a calculation within the example is resolved using a method of complex approximation.

A Survey Of The Taniyama-Shimura Conjecture, Kady Schneiter

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Perhaps the most famous problem in all of mathematics is the theorem that states that the equation aⁿ + bⁿ = cⁿ has no non-trivial solutions for integers a, b, and c, and n ≥ 2. This theorem was proposed by a seventeenth century French mathematician named Pierre de Fermat. Though the theorem is easy to understand, the proof has been elusive. Over the past 350 years many mathematicians have attempted to prove Fermat's theorem. They have used a variety of methods and many have been successful in proving the theorem in specific cases. …

The Poincaré Conjecture, Joseph D. Peck

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The central theme for this paper is provided by the following three statements:

(1) Every compact connected 1-manifold is S1.

(2) Every compact connected simply connected 2-manifold is S2.

(3) Every compact connected simply connected 3-manifold is S3.

We provide proofs of statements (1) and (2). The veracity of the third statement, the Poincaré Conjecture, has not been determined. It is known that should a counter-example exist it can be found by removing from S3 a finite collect ion of solid tori and sewing them back differently. We show that it is not possible to find a counterexample by removing …