Solving quadratic congruence equations using a pseudo-random (Tonelli-Shanks) algorithm is discussed. We give several examples and many workable exercises. Introduction to Tonelli-Shanks Algorithm The Tonell-Shanks algorithm (sometimes called the RESSOL algorithm) is used within modular arithmetic where $a$ is a quadratic residue (mod $p$), and $p$ is an odd prime. Tonelli–Shanks cannot be used for … Read More

First we discuss transforming and solving quadratic congruence equations. We then study quadratic residues using the Legendre symbol. Euler’s and Gauss’s Criterions are motivated and then the infamous Law of Quadratic Reciprocity is understood. General Quadratic Congruence Consider the general quadratic congruence, $$a x^2+b x+c\equiv 0 \pmod{p}$$ where $p$ is an odd prime and $(a,p)=1.$ … Read More

We prove several properties of Euler’s Totient Function and give many examples. We also discuss solving functional equations and reduced residue systems. One of Euler’s most important theorems is then demonstrated and proven. Introduction to Euler’s Totient Function Definition. For each integer $n>1,$ let $\phi (n)$ denote the number of positive integers less than $n$ … Read More

Wilson’s theorem, its converse, and Fermat’s theorem are discussed. We motivate each proof through example and careful write out the proof of each theorem. Several examples of their use are given. Wilson’s Theorem This theorem is named after one of Edward Waring‘s students, John Wilson. But actually, Wilson only observed the result to be true … Read More

This definitive guide covers proofs, examples, algorithms, applications, and history of the Chinese Remainder Theorem. It also includes links to additional resources such as online articles, courses, books, and tutors to help students learn from a variety of sources. Professionals can also use these resources to increase their knowledge of the field or help structure courses for their students.

We discuss two applications of congruence problems. How to develop a divisibility test, emphasizing theory as well as usability. We then discuss the infamous Days of the Week problem. Applications of Congruence – Divisibility Tests Our first application of congruence is a collection of theorems which help determine divisibility of an integer with another. Divisibility … Read More

The idea behind solving polynomial congruence equations is that we can reduce a congruence equation to an equivalent system of congruence equations using prime factorization. We then 1) solve each equation modulo a prime number (by brute force), 2) use Hensel’s Lifting theorem, and then 3) piece together the solutions using the Chinese Remainder Theorem. … Read More

We study the solvability of linear congruence equations and practice solving them. We also discuss incongruent solutions and characterize solvability using inverses. Then we place significance on using the Euclidean algorithm, solving linear Diophantine methods, and importantly, on using an ad hoc method. Introduction to Linear Congruences Linear congruences are the main object of discussion … Read More

We discuss congruence theorems on the integers by proving several elementary lemmas. Modular arithmetic and least positive residues is also discussed. Introduction to Congruence Theorems A modern treatment of congruences was introduced by Carl Friedrich Gauss. Congruence, or modular arithmetic, arises naturally in common everyday situations. For example, odometers usually work modulo 100,000 and utility … Read More

We solve linear Diophantine equations over the integers. In the two variable case, we provide a complete solution using the Euclidean Algorithm. Also, we will discuss the multi-variable case and provide examples. We will take note of the fact that a Diophantine Equation of the linear kind seeks to equate the sum of two or … Read More

First we characterize prime numbers in terms of divisibility. We then use this characterization to prove the Fundamental Theorem of Arithmetic –that every positive integer has a unique factorization into a product of primes. We also discuss greatest common divisors and least common multiples. Finally, we will also provide the proof to the theorem, which … Read More

The Euclidean Algorithm is an ancient method for finding the greatest common divisors (gcd) of two integers. The algorithm also generates the information necessary to write the gcd as a linear combination of the given integers. The Euclidean Algorithm is proven using previously established lemmas. Finally, emphasis is placed on its proof and examples of … Read More

Number Theory has a long and exciting history. To help understand what Number Theory is all about, in this article, we describe a few basic ideas of Number Theory. From divisibility and mathematical induction to Euler’s theorem and solving polynomial congruence equations, Number Theory can be both highly practical and applicable yet also extremely difficult … Read More

Briana Bonfiglio is a newspaper reporter based in Long Island, New York, and holds a bachelor’s degree in journalism from the State University of New York at New Paltz. She has written for several publications and websites, including Chronogram, The New Paltz Oracle, and Direct Knowledge. She is currently a reporter for Long Island Herald Community Newspapers.

Nian is an anesthesiology resident in New York. She is also an entrepreneur and specializes in helping early-career physicians start their online businesses in residency. She helps residents overcome burnout by empowering them to build meaningful and profitable businesses out of their passions and hobbies, to a level where they have the freedom and choice to leave clinical medicine.

This article is about Direct Knowledge contributor Chelsea Turner. Although her primary work is in college counseling, she is an avid writer. Although her area of expertise is the humanities, particularly language and communication, Chelsea has published articles on an array of topics. This biography delves into Chelsea’s background, education, and professional experience.

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Taylor Bauer is a technical writer and digital marketing professional. Read about his educational background and the work he finished during undergraduate and graduate school. Explore areas of his expertise and what Taylor does now in his career. He is a contributor to Direct Knowledge and is passionate about providing educational resources to all who want to learn.

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Megan Matheney is a professional technical writer specializing in material relating to Earth sciences, environmental economics, and developmental economics. Her range of knowledge comes from a B.S. in Geophysics from the University of Texas at Austin. And from an M.S. in Environment and Sustainable Development from the University of Glasgow.