Number Theory Commons^™

Paley Graphs, Prime Graphs, And Crossword Puzzles, Robert D. Jacobs Jr.

Theses and Dissertations

In this paper, we will talk about many different mathematical concepts. We will prove theorems about Paley graphs, prime graphs, and crossword puzzles. It will be very fun.

The results in the section about Paley graphs include structure theorems about the subgraph induced by the quadratic residues, the subgraph induced by the non-residues and a few related subgraphs. The main is to better understand the “independence structure” of the Paley graph itself. No good upper bound on the independence number of Paley graphs is known. Theorems about these subgraphs, and various counts aim at future improvement of upper bounds for …

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.

One Formula For Non-Prime Numbers: Motivations And Characteristics, Mahmoud Mansour, Kamal Hassan Prof.

Basic Science Engineering

Primes are essential for computer encryption and cryptography, as they are fundamental units of whole numbers and are of the highest importance due to their mathematical qualities. However, identifying a pattern of primes is not easy. Thinking in a different way may get benefits, by considering the opposite side of the problem which means focusing on non-prime numbers. Recently, researchers introduced, the pattern of non-primes in two maximal sets while in this paper, non-primes are presented in one formula. Getting one-way formula for non-primes may pave the way for further applications based on the idea of primes.

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.

On The Mersenne Prime Numbers, Julia Vanlandingham

Undergraduate Honors Thesis Projects

The prime numbers have been an important field of research for thousands of years and are intertwined with most other fields of mathematics. One topic that has piqued the interest of mathematicians young and old is the Mersenne prime numbers, which have applications in many mathematics and computer science fields. The Mersenne primes get a lot of attention because there is not much known about them. However, we do have a very simple primality test for Mersenne numbers, which is why the largest currently known primes are Mersenne primes. These primes are also very closely related to another class of …

Numbers In Base B That Generate Primes With Help The Luhn Function Of Order Ω, Florentin Smarandache, Octavian Cira

Branch Mathematics and Statistics Faculty and Staff Publications

We put the problem to determine the sets of integers in base b ≥ 2 that generate primes with using a function.

Smarandache Type Functions Obtained By Duality, Florentin Smarandache, C Dumitrescu, N Varlan, St Zamfir, E Radescu, N Radescu

Branch Mathematics and Statistics Faculty and Staff Publications

we extend the Smarandache function from the set N* of positive integers to the set Q of rationals