Number Theory Commons^™

Arithmetic Theorems, Demonstrated By A New Method: An English Translation Of E271, Brendon Lasell

Euleriana

After defending in general terms the value of arithmetic demonstrations, where, according to Euler, the force of genius shines more brightly than in any other kind of demonstration, he goes on to let the force of his own genius shine forth in demonstrating the fact that, for any pair of relatively prime numbers a and N, a^ϕ(N) − 1 is always divisible by N, where we anachronistically denote by ϕ(N) the number of numbers less than N that are relatively prime to it. He approaches this theorem by considering the remainders that result when the numbers in an arithmetic …

Gauss Composition And Orthogonal Modular Forms On Binary Lattices, Haochen Wu

Dartmouth College Ph.D Dissertations

We revisit Gauss composition over a general base scheme, with a focus on orthogonal groups. We show that the Clifford and norm functors provide a discriminant-preserving equivalence of categories between binary quadratic modules and pseudoregular modules over quadratic algebras. This perspective synthesizes the constructions of Kneser and Wood, reconciling algebraic and geometric approaches and clarifying the role of orientations and the natural emergence of narrow class groups.

As an application, we restrict to lattices and show that binary orthogonal eigenforms correspond to Hecke characters. Using theta series, we show the explicit connection between Hilbert modular forms and orthogonal modular forms …

When A Sum Of Cubes Equals The Square Of The Sum, Jessica M. Aguilar

Electronic Theses, Projects, and Dissertations

This thesis investigates extensions and structural generalizations of the classical identity \[ \sum_{k=1}^{n} k^3 = \left( \sum_{k=1}^{n} k \right)^2, \] traditionally attributed to Nicomachus of Gerasa. Despite its simple look, this cube - square identity reveals connections between combinatorics, multiplicative number theory, and Diophantine equations.

We begin by presenting an expanded combinatorial proof of the identity based on Stein’s rectangle - counting argument, clarifying the geometric structure underlying the formula. We then establish a multiplicative analogue using Euler’s divisor-counting function \( \tau(n) \), proving that \[ \sum_{d \mid n} \tau(d)^3 = \left( \sum_{d \mid n} \tau(d) \right)^2, \] thereby extending …

Universal Centralizers, Morita Abelianization, And Wonderful Models In Lie Theory, Peter Crooks

Funded Research Records

No abstract provided.

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.

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 …

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.

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, …

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.

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.

Nessie Notation: A New Tool In Sequential Substitution Systems And Graph Theory For Summarizing Concatenations, Colton Davis

Student Research

While doing research looking for ways to categorize causal networks generated by Sequential Substitution Systems, I created a new notation to compactly summarize concatenations of integers or strings of integers, including infinite sequences of these, in the same way that sums, products, and unions of sets can be summarized. Using my method, any sequence of integers or strings of integers with a closed-form iterative pattern can be compactly summarized in just one line of mathematical notation, including graphs generated by Sequential Substitution Systems, many Primitive Pythagorean Triplets, and various Lucas sequences including the Fibonacci sequence and the sequence of square …

Structure Of Number Theoretic Graphs, Lee Trent

Mathematical Sciences Technical Reports (MSTR)

The tools of graph theory can be used to investigate the structure
imposed on the integers by various relations. Here we investigate two
kinds of graphs. The first, a square product graph, takes for its vertices
the integers 1 through n, and draws edges between numbers whose product
is a square. The second, a square product graph, has the same vertex set,
and draws edges between numbers whose sum is a square.
We investigate the structure of these graphs. For square product
graphs, we provide a rather complete characterization of their structure as
a union of disjoint complete graphs. For …

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell

Theses

This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued fractions without the …

The Generalized Riemann Hypothesis And Applications To Primality Testing, Peter Hall

University Scholar Projects

The Riemann Hypothesis, posed in 1859 by Bernhard Riemann, is about zeros
of the Riemann zeta-function in the complex plane. The zeta-function can be repre-
sented as a sum over positive integers n of terms 1/ns when s is a complex number
with real part greater than 1. It may also be represented in this region as a prod-
uct over the primes called an Euler product. These definitions of the zeta-function
allow us to find other representations that are valid in more of the complex plane,
including a product representation over its zeros. The Riemann Hypothesis says that
all …

Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang

Electronic Theses and Dissertations

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …

A Case Study On Hooley's Conditional Proof Of Artin's Primitive Root Conjecture, Shalome Kurian

Rose-Hulman Undergraduate Mathematics Journal

Artin’s Primitive Root Conjecture represents one of many famous problems in elementary number theory that has resisted complete solution thus far. Significant progress was made in 1967, when Christopher Hooley published a conditional proof of the conjecture under the assumption of a certain case of the Generalised Riemann Hypothesis. In this survey we present a description of the conjecture and the underlying algebraic theory, and provide a detailed account of Hooley’s proof which is intended to be accessible to those with only undergraduate level knowledge. We also discuss a result concerning the qx+1 problem, whose proof requires similar techniques to …

On The Local Theory Of Profinite Groups, Mohammad Shatnawi

Dissertations

Let G be a finite group, and H be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of H. The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group G into H/H^' an abelian group where H is a subgroup of G and H' is the derived group of H. One important first application is Burnside’s normal p-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it. …

The Name Tag Problem, Christian Carley

Rose-Hulman Undergraduate Mathematics Journal

The Name Tag Problem is a thought experiment that, when formalized, serves as an introduction to the concept of an orthomorphism of $\Zn$. Orthomorphisms are a type of group permutation and their graphs are used to construct mutually orthogonal Latin squares, affine planes and other objects. This paper walks through the formalization of the Name Tag Problem and its linear solutions, which center around modular arithmetic. The characterization of which linear mappings give rise to these solutions developed in this paper can be used to calculate the exact number of linear orthomorphisms for any additive group Z/nZ, which is demonstrated …

An In-Depth Look At P-Adic Numbers, Xiaona Zhou

Publications and Research

In this study, we consider $p$-adic numbers. We will also study the $p$-adic norm representation of real number, which is defined as $\mathbb{Q}_p = \{\sum_{j=m}^{\infty }a_j p^j: a_j \in \mathbb{D}_p, m\in\mathbb{Z}, a_m\neq 0\} \cup \{0\}$, where $p$ is a prime number. We explore properties of the $p$-adics by using examples. In particular, we will show that $\sqrt{6},i \in \mathbb{Q}_5$ and $\sqrt{2} \in \mathbb{Q}_7 $. $p$-adic numbers have a wide range of applicationsnin fields such as string theory, quantum mechanics, and transportation in porous disordered media in geology.

On The Equality Case Of The Ramanujan Conjecture For Hilbert Modular Forms, Liubomir Chiriac

Mathematics and Statistics Faculty Publications and Presentations

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations π on GL(2) asserts that |av(π)| ≤ 2. We prove that this inequality is strict if π is generated by a CM Hilbert modular form of parallel weight two and v is a finite place of degree one. Equivalently, the Satake parameters of πv are necessarily distinct. We also give examples where the equality case does occur for primes of degree two.

Inquiry In Inquiry: A Classification Of The Learning Theories Underlying Inquiry-Based Undergraduate Number Theory Texts, Rebecca L. Butler

Honors Projects

While undergraduate inquiry-based texts in number theory share similar approaches with respect to learning as the embodiment of professional practice, this does not entail that these texts all operate from the same fundamental understanding of what it means to learn mathematics. In this paper, the instructional design of several texts of the aforementioned types are analyzed to assess the theory of learning under which they operate. From this understanding of the different theories of learning employed in an inquiry-based mathematical setting, one can come to understand the popular model of what it is to learn number theory in a meaningful …

Vector Partitions, Jennifer French

Electronic Theses and Dissertations

Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The primary purpose …

Pgl2(FL) Number Fields With Rational Companion Forms, David P. Roberts

Mathematics Publications

We give a list of PGL₂(F_l) number fields for ℓ ≥ 11 which have rational companion forms. Our list has fifty-three fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.