Mathematics Commons^™

Algorithms For Order Statistics In Farey Sequences: A Computational Study, Connor Weyers

School of Computing: Dissertations, Theses, and Student Research

Farey sequences are the sets of irreducible fractions in increasing order with denominator less or equal to some integer n. They are a well-known concept in number theory problems and are related to many other concepts in number theory including integer factoring, Fibonacci sequences, and Riemann’s Zeta function. In this paper, we investigate some known algorithms to solve certain problems in Farey sequences from a computational perspective. In particular, we implement established algorithms that have not been previously implemented with the goal of creating a package that can be used more broadly. We also develop a new algorithm for rational …

Galoistheory And The Arithmetic-Geometric Series, Daniel Vargas

HMC Senior Theses

Motivated by classical works of Gauss and Euler on the AGM, Ono and his

collaborators Griffin et al. (2023); McSpirit and Ono (2023) have investigated

the union of AGM sequences over finite fields 𝔽𝑞, where 𝑞 ≡3 mod 4. A

recent preprint Kayath et al. (2024) extends some of their results to all finite

fields with odd characteristic. We refine these works when 𝑞≡5 mod 8. In

particular, we explicitly determine the components of these graphs and their

total population. We also use Galois-theoretic results to make progress in

the search for cycles over finite fields with odd characteristic.

Diophantine Avoidance, Number Fields, And Quadratic Forms, Sehun Jeong

CGU Theses & Dissertations

Diophantine avoidance has been studied by several authors in recent years. This term refers to effective results on existence of points of bounded size (where size is measured by norm or height, depending on the context) in a given algebraic set avoiding some specified subsets. The application of avoidance conditions allows to understand how ``well distributed" are points of bounded size in a given set. If it is possible to find them outside of some prescribed collection of subsets of the set in question, then it suggests that they are evenly distributed, in some appropriate sense. Our first result investigates …

Golden Spirals Everywhere?, John Adam

Mathematics & Statistics Faculty Publications

The article explores different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It discusses the misconception that golden rectangles and spirals can be found in various natural and man-made objects, emphasizing the importance of understanding the properties of logarithmic spirals. The text provides mathematical equations for logarithmic spirals and poses questions for readers to explore the concept further. The author, John Adam, invites readers to engage in Fermi Questions and submit ideas for consideration.

Golden Spirals Everywhere? Solutions For Fermi Questions, January 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It addresses the misconception that golden rectangles and spirals can be found in various natural and man-made structures, emphasizing the importance of understanding the properties of logarithmic spirals. The article provides mathematical explanations and solutions for questions related to pitch angles and self-similarity in logarithmic spirals, using examples like the nautilus shell and an ammonite-like stone. It concludes by referencing additional sources for further exploration of the golden ratio and golden spiral myths.

Limit Theorems For L-Functions In Analytic Number Theory, Asher Roberts

Dissertations, Theses, and Capstone Projects

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using …

New Algorithms For The Multiplication Table Problem, Evan Blom

Undergraduate Honors Thesis Collection

In 1955, Paul Erdős initiated the study of a function that counts the number of distinct integers in an (n × n) multiplication table. That is, he studied M(n) = |{i · j, 1 ≤ i, j ≤ n}|. Much research has been done in regards to both asymptotic and exact approximations of M(n) for increasingly large values of n. Recently, Brent et. al. investigated the algorithmic cost in computing this function. Instead of computing M(n) directly, their approach was to compute it incrementally. That is, given M(n−1), they could quickly compute M(n) using another function δ(n) to count the …

Research On Arithmetic, Erik R. Tou

Euleriana

In this English translation, some of Joseph-Louis Lagrange's early number theory is presented. Here, he laid out a theory of binary quadratic forms with special attention to the representation problem: determining those integers which may be represented by a given form, and cataloguing the possible forms of their divisors.

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

A Spiral Workbook For Discrete Mathematics 2nd Edition, Harris Kwong

Milne Open Textbooks

This updated text covers the standard topics in a sophomore-level course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions, relations, and elementary combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a final polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: many topics are revisited multiple times, sometimes from a different perspective or at a higher level of complexity. The goal is …

Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

Theses and Dissertations--Mathematics

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.

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, …

Solution Of The Diophantine Equation (Maa+Nbb)=Cd(Mcc+Ndd) Using Rational Numbers, Georg Ehlers

Euleriana

This paper (E716) was published in Nova acta Academiae scientiarum imperialis petropolitanae, Volume 13 (1795/96), pp. 45-63. It was also included in Commentationes Arithmeticae, Volume II, as Number LXVIII, pp. 281-293 (E791). Euler starts with Fermat's Last Theorem and mentions the proofs for the cases n=3 and n=4 which he had completed himself earlier. He then moves on to make the sum of powers conjecture, which was later disproved in the second half of the 20th century. In this context he discusses his discovery of 134^4+133^4=158^4+59^4, which he calls unexpected. Euler derives the title equation from A^4+B^4=C^4+D^4, generalizing it to …

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

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.

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

Euleriana

A survey of two translations posted to the Euler Archive in 2022.

Euler's Anticipations, Christopher Goff, Erik Tou

Euleriana

Welcome to Volume 3 of Euleriana. This issue highlights occasions where Euler's work anticipated future results from other others, sometimes by decades or even centuries!

Unsolved Haiku, Scott W. Williams

Journal of Humanistic Mathematics

This poem describes the still unsolved 1937 conjecture of Lloyd Collatz: Do repeated applications of the algorithm described yield the number 1?

The Genesis Of A Theorem, Osvaldo Marrero

Journal of Humanistic Mathematics

We present the story of a theorem's conception and birth. The tale begins with the circumstances in which the idea sprouted; then is the question's origin; next comes the preliminary investigation, which led to the conjecture and the proof; finally, we state the theorem. Our discussion is accessible to anyone who knows mathematical induction. Therefore, this material can be used for instruction in a variety of courses. In particular, this story may be used in undergraduate courses as an example of how mathematicians do research. As a bonus, the proof by induction is not of the simplest kind, because it …

Lattice Extensions And Zeros Of Multilinear Polynomials, Maxwell Forst

CGU Theses & Dissertations

We treat several problems related to the existence of lattice extensions preserving certain geometric properties and small-height zeros of various multilinear polynomials. An extension of a Euclidean lattice $L_1$ is a lattice $L_2$ of higher rank containing $L_1$ so that the intersection of $L_2$ with the subspace spanned by $L_1$ is equal to $L_1$. Our first result provides a counting estimate on the number of ways a primitive collection of vectors in a lattice can be extended to a basis for this lattice. Next, we discuss the existence of lattice extensions with controlled determinant, successive minima and covering radius. In …

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 …

Meertens Number And Its Variations, Chai Wah Wu

Communications on Number Theory and Combinatorial Theory

In 1998, Bird introduced Meertens numbers as numbers that are invariant under a map similar to the Gödel encoding. In base 10, the only known Meertens number is 81312000. We look at some properties of Meertens numbers and consider variations of this concept. In particular, we consider variations of Meertens numbers where there is a finite time algorithm to decide whether such numbers exist, exhibit infinite families of these variations and provide bounds on parameters needed for their existence.

Squate, Tom Blackford

Journal of Humanistic Mathematics

This is the story of a middle school student who befriends an irrational number, the square root of eight.