Mikhail Ivanov
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Math 467: Introduction to Number Theory
Have taught Fall 2026, Fall 2023
- 2023/ Syllabus
- 2023/ Exam 1, Solutions
- 2023/ Exam 2, Solutions
- 2023/ Exam 3, Solutions
- 2023/ Exam 4, Solutions
- 2023/ Topics: Mathematical Induction, the Binomial Theorem, the Division Algorithm, the Greatest Common Divisor,
the Euclidean Algorithm, the Diophantine Equation $ax + by = c$, the Fundamental Theorem of Arithmetic, the Sieve of Eratosthenes,
the Theory of Congruences, Binary and Decimal Representations of Integers, Linear Congruences, the Chinese Remainder Theorem,
Fermat's Little Theorem, Pseudoprimes, the Sum and Number of Divisors, The Mobius Inversion Formula, the Greatest Integer Function,
Euler's Phi-Function, Euler's Theorem, The Pythagorean Triples, Fermat's Last Theorem for $n=4$, the Fibonacci Sequence,
the Fibonacci Sequence modulo n.
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Math 541 Modern Algebra I
Have taught Fall 2026, Fall 2025
- 2025/ Syllabus
- 2025/ Midterm 1, Solutions
- 2025/ Midterm 2, Solutions
- 2025/ Final , Solutions
- 2025/ Book: David Dummit & Richard Foote. Abstract Algebra. Wiley. 3th ed.
- 2025/ Content
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Math/Comp Sci/E C E 435: Introduction to Cryptography
Have taught Spring 2026, Spring 2024
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Math 567 Modern Number Theory
Have taught Spring 2025
- 2025/ Syllabus
- 2025/ Midterm 1, Solutions
- 2025/ Midterm 2, Solutions
- 2025/ Final, Solutions
- 2025/ Book: William Stein "Elementary Number Theory: Primes, Congruences, and Secrets" + additional resourses
- 2025/ Content $\newcommand{\Z}{\mathbb Z}$
- Euclidean division and $\gcd$
- Fundamental Theorem of Arithmetic in $\Z$
- Congruences
- The Euler, Fermat, Wilson Theorems
- Chinese Remainder theorem
- Multiplicative functions
- Linear Diophantine Equations
- Fast powers and pseudoprimes
- The group of units of $\Z/n\Z$, Primitive roots
- RSA
- DHKE
- Quadratic Reciprocity
- Elementary proof of Quadratic Reciprocity Law
- Gauss sums
- Quadratic Reciprocity using Gauss sums
- Continued fractions
- Convergence of partial convergents
- The Continued Fraction of e
- Quadratic Irrationals
- Rational approximations
- Sums of Two Squares
- Pell’s equation
- The Pythagorean triples
- Elliptic curves
- Examples of the group law
- Elliptic curves over different fields
- Pollard's $(p-1)$ method and Lenstra’s elliptic curve factorization
- ECDHKE and El-Gamal
- Congruent Numbers
- FLT for $n = 4$
- Euclidean domains. PID. UFD
- Arithmetic in $\Z[i]$
- Another quadraic rings
- Arithmetic in $\Z[ω]$
- FLT for $n=3$
- Bertand Postulate
- Weak prime number theorem
- ζ-function
- $L$-functions and Dirichlet’s Theorem
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Math/Stat 431 Introduction to Probability
Have taught Spring 2025 (2 sections), Fall 2024 (2 sections), Fall 2021
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Small group tutoring for Math 222 "Calculus II"
- Math Learning Center
- Content of Fall 2024
- (Review) Derivatives, Integration, Trigonometric and Inverse Trigonometric Functions. Please come to our review sessions
- (7.1) Integration by Parts
- (7.2) Trigonometric Integrals
- (7.3) Trigonometric Substitutions
- (7.4) Integration of Rational Functions
- (7.8) Improper Integrals
- (9.1, 9.2) Modeling with Differential Equations, Direction Fields and Euler's Method
- (Review) Midterm 1. Please come to our review sessions
- (9.3) Separable Equations
- (9.5) Linear Equations
- (10.1, 10.2) Curves Defined by Parametric Equations, Calculus with Parametric Curves
- (10.3) Polar Coordinates
- (Review) Limits. Please come to our review sessions
- (11.1) Sequences
- (11.2) Series
- (11.3) The Integral Test and Estimates of Sums
- (11.4) The Comparison Tests
- (11.5) Alternating Series
- (Review) Midterm 2. Please come to our review sessions
- (11.6) Absolute Convergence and the Ratio and Root Tests
- (11.8) Power Series
- (11.9) Representations of Functions as Power Series
- (11.10) Taylor and Maclaurin Series
- (12.1, 12.2) Three-Dimensional Coordinate Systems, Vectors
- (12.3) The Dot Product
- (12.4) The Cross Product
- (12.5) Equations of Lines and Planes
- (Review) Final Exam. Please come to our review sessions
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Math/Comp Sci 240: Introduction to Discrete Mathematics
Have taught Spring 2024
- 2024/ Syllabus
- 2024/ Midterm 1, Solutions
- 2024/ Midterm 2, Solutions
- 2024/ Final, Solutions
- 2024/ Topics: Integer properties, Logic of Propositions, Logic of Predicates. Proofs, Sets, Floor and Ceiling Functions,
Injection, Surjection, Bijection, Inverse Functions, Asymptotic Analysis (Big-Oh), Algorithm Analysis, Finite-state machines,
Sequences, Summation. Induction, Recursive definitions, Structural Induction, Iterative and Recursive algorithms, Program Correctness,
Master Theorem, Recurrence Relations, Relations on sets, Equivalence relations, Partial orders, Enumeration, Pengionhole Priciple,
Binomial Coefficients, Graphs, Trees.
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Math/Comp Sci/Stat 475 Introduction to Combinatorics
Have taught Fall 2023, Fall 2021, Spring 2021, Fall 2020
- 2023/ Syllabus
- 2023/ Midterm 1, Solutions
- 2023/ Midterm 2, Solutions
- 2023/ Final, Solutions
- 2023/ Topics: Basic counting principles, k-permutations and k-combinations of sets,
Permutations and combinationcs of multisets, Probability, Pigeonhole Principle, Inversions, Lexicographic order,
Reflected Gray code, Binary Relations, Partial orders, Equivalence relations, Binomial Coefficients, Sperner Theorem,
Multinomial Theorem, Newton's Binomial Theorem for integer exponents, Inclusion-Exclusion Formula and Applications,
Forbidden positions, Fibonacci sequence, Generating Functions, Exponential Generating Functions, Linear recurrence relations,
Stirling numbers, Partition numbers, Basic Properties of Graphs, Paths and Connectivity, Eulerian Trails,
Hamiltonian paths and cycles, Bipartite graphs, Graph Colorings, Ramsey Theory, Permutations Groups, Burnside's Theorem,
Polya Counting.
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Math/Hist Sci 473 History of Mathematics
Have taught Spring 2023
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Math 222 Calculus and Analytic Geometry II
Have taught Spring 2023, Spring 2022
- 2023/ Syllabus
- Ask our Calculus coordinator about materials
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Math 211 Calculus
Have taught Fall 2022
- 2022/ Syllabus
- Ask our Calculus Coordinator about materials
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Math 461 College Geometry
Have taught Spring 2022, Spring 2021