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Jan 15, 2025
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MA 388 - Introduction to Number Theory3 lecture hours 0 lab hours 3 credits
Course Description
Number theory is primarily concerned with the properties of the integers. While the subject has long been thought of as quintessentially “pure” mathematics, recent developments in fields such as cryptography have renewed interest in it. Topics include: mathematical induction; divisibility and primes; the Euclidean algorithm; linear Diophantine equations; modular arithmetic; primality testing; continued fractions. (prereq: MA 231 )
Course Learning Outcomes
Upon successful completion of this course, the student will be able to:
• Write elementary proofs
• Use the principle of mathematical induction
• Apply the Euclidean algorithm and solve linear Diophantine equations
• Perform modular arithmetic
• Apply Fermat’s Little Theorem and Euler’s Theorem
• Understand the distribution of the prime numbers
• Test for primality of integers
• Find continued fraction expressions for real numbers (optional)
• Understand the RSA encryption algorithm
• Use Quadratic Reciprocity to compute Legendre symbols
Prerequisites by Topic
• No prerequisites by topic appended
Course Topics
• Introduction to number theory, mathematical proof, and induction (4 classes)
• Euclidean algorithm, divisibility, the GCD, and linear Diophantine equations (4 classes)
• Fundamental Theorem of Arithmetic (1 class)
• Congruences and Fermat’s Little Theorem. (3 classes)
• The Phi Function and Euler’s Theorem (2 classes)
• Chinese Remainder Theorem (1 class)
• Distribution of Primes; Primality testing. (2 classes)
• Successive squaring, k-th roots, and RSA (3 classes)
• Primitive Roots and Discrete Logarithms (2 classes)
• Quadratic Reciprocity (3 classes)
• Reviews and exams (5 classes)
Coordinator
Anthony van Groningen
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