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Jeffrey Ehme – PRIMUS, 2024
The Miller-Rabin test is a useful probabilistic method for finding large primes. In this paper, we explain the method in detail and give three variations on this test. These variations were originally developed as student projects to supplement a course in error correcting codes and cryptography.
Descriptors: Probability, Numbers, Coding, Algorithms
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Phillips, Matthew; Robb, Kayla; Shipman, Barbara A. – PRIMUS, 2023
In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and…
Descriptors: Capstone Experiences, Algebra, Mathematics Education, Mathematics Instruction
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Karaali, Gizem; Yih, Samuel – PRIMUS, 2020
When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number…
Descriptors: Mathematics Instruction, College Mathematics, Algebra, Numbers
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Dana C. Ernst; Jeffrey Slye – PRIMUS, 2024
The algebraic group Spin[subscript 3 × 3] arises from spinning collections of the numbers 1-9 on a 3×3 game board. The authors have been using this group, as well as a corresponding online application, to introduce undergraduate students to core concepts in group theory. We discuss the benefits of using this deceptively simple, toy-like puzzle in…
Descriptors: Algebra, Numbers, Computer Oriented Programs, Undergraduate Students
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Herrera, Christine A.; McCabe, Terrance; Strictland, Sharon; White, Alexander – PRIMUS, 2018
In an undergraduate analysis course taught by one of the authors, three prompts are regularly given: (i) What do we know? (ii) What do we need to show? (iii) Let's draw a picture. We focus on the third prompt and its role in helping students develop their confidence in learning how to construct proofs. Specific examples of visual models and their…
Descriptors: Mathematics Instruction, Mathematical Logic, Validity, Mathematics Skills
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Garcia, Stephan Ramon – PRIMUS, 2017
A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by…
Descriptors: Algebra, Mathematics Instruction, Numbers, College Mathematics
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Das, Tushar; George, Whitney; Warnberg, Nathan – PRIMUS, 2020
Mathematics educators view the equals sign as a bidirectional relation symbol, but the authors have observed that students might not have such flexibility in their understanding of the equals sign. The authors have observed that students have a hard time viewing mathematical properties (for example, log (MN) = log (M) log (N)) with bidirectional…
Descriptors: Mathematics Education, Mathematical Concepts, Symbols (Mathematics), College Mathematics
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Katrina Palmer; William Bauldry; Michael J. Bossé; Jaehee Post – PRIMUS, 2022
Most any students can explain the meaning of "a[superscript b]", for "a" [element-of] [set of real numbers] and for "b" [element-of] [set of integers]. And some students may be able to explain the meaning of "(a + bi)[superscript c]," for "a, b" [element-of] [set of real numbers] and for…
Descriptors: Mathematics Instruction, Mathematical Concepts, Secondary School Mathematics, College Mathematics
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Coggins, Porter E., III; Glatzer, Tim – PRIMUS, 2020
We present an algorithm for a matrix-based Enigma-type encoder based on a variation of the Hill Cipher as an application of 2 × 2 matrices. In particular, students will use vector addition and 2 × 2 matrix multiplication by column vectors to simulate a matrix version of the German Enigma Encoding Machine as a basic example of cryptography. The…
Descriptors: Mathematics Instruction, Matrices, Technology, Addition
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Smith, Michael D. – PRIMUS, 2016
The purpose of this article is to share a new approach for introducing students to the definition and standard examples of Abelian groups. The definition of an Abelian group is revised to include six axioms. A bullseye provides a way to visualize elementary examples and non-examples of Abelian groups. An activity based on the game of Jenga is used…
Descriptors: Undergraduate Study, College Mathematics, Games, Manipulative Materials
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Sachs, Robert – PRIMUS, 2017
A new transition course centered on complex topics would help in revitalizing complex analysis in two ways: first, provide early exposure to complex functions, sparking greater interest in the complex analysis course; second, create extra time in the complex analysis course by eliminating the "complex precalculus" part of the course. In…
Descriptors: Mathematics Instruction, Undergraduate Study, Validity, Mathematical Logic
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Bosse, Michael J.; Ries, Heather; Chandler, Kayla – PRIMUS, 2012
Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…
Descriptors: Algebra, Number Systems, Secondary School Mathematics, Elementary School Mathematics
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Aydin, Nuh – PRIMUS, 2009
The theory of error-correcting codes and cryptography are two relatively recent applications of mathematics to information and communication systems. The mathematical tools used in these fields generally come from algebra, elementary number theory, and combinatorics, including concepts from computational complexity. It is possible to introduce the…
Descriptors: Mathematics Curriculum, Computer Science, Number Concepts, Liberal Arts