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Articles 1 - 8 of 8
Full-Text Articles in Number Theory
Further Generalizations Of Happy Numbers, E. Simonton Williams
Further Generalizations Of Happy Numbers, E. Simonton Williams
Rose-Hulman Undergraduate Mathematics Journal
A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, …
Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov
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.
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Rose-Hulman Undergraduate Mathematics Journal
A magic square is a square grid of numbers where each row, column, and long diagonal has the same sum (called the magic sum). An open problem popularized by Martin Gardner asks whether there exists a 3×3 magic square of distinct positive square numbers. In this paper, we expand on existing results about the prime factors of elements of such a square, and then provide a full list of the ways a prime factor could appear in one. We also suggest a separate possible computational approach based on the prime signature of the center entry of the square.
Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Doctoral Dissertations
We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called "Ramanujan congruences" for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits.
Number Theoretic Arithmetic Functions And Dirichlet Series, Ivan V. Morozov
Number Theoretic Arithmetic Functions And Dirichlet Series, Ivan V. Morozov
Publications and Research
In this study, we will study number theoretic functions and their associated Dirichlet series. This study lay the foundation for deep research that has applications in cryptography and theoretical studies. Our work will expand known results and venture into the complex plane.
Euler Archive Spotlight, Erik R. Tou
Euler Archive Spotlight, Erik R. Tou
Euleriana
A survey of two translations posted to the Euler Archive in 2022.
Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer
Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer
Senior Projects Spring 2023
Over the course of history, western music has created a unique mathematical problem for itself. From acoustics, we know that two notes sound good together when they are related by simple ratios consisting of low primes. The problem arises when we try to build a finite set of pitches, like the 12 notes on a piano, that are all related by such ratios. We approach the problem by laying out definitions and axioms that seek to identify and generalize desirable properties. We can then apply these ideas to a broadened algebraic framework. Rings in which low prime integers can be …
Elliptic Functions And Iterative Algorithms For Π, Eduardo Jose Evans
Elliptic Functions And Iterative Algorithms For Π, Eduardo Jose Evans
UNF Graduate Theses and Dissertations
Preliminary identities in the theory of basic hypergeometric series, or `q-series', are proven. These include q-analogues of the exponential function, which lead to a fairly simple proof of Jacobi's celebrated triple product identity due to Andrews. The Dedekind eta function is introduced and a few identities of it derived. Euler's pentagonal number theorem is shown as a special case of Ramanujan's theta function and Watson's quintuple product identity is proved in a manner given by Carlitz and Subbarao. The Jacobian theta functions are introduced as special kinds of basic hypergeometric series and various relations between them derived using the triple …