Study properties of integers and number systems
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117
Sub Topics
174
MCQs
76
MCOs
129
True/False
69
Fill Blanks
24
Rearrange
67
Matching
31
Comprehensions
70
Flashcard Decks
Curriculum
What You'll Learn
01 Introduction to Number Theory 3 topics
1 Historical Development
- Ancient Number Theory
- Greek Contributions
- Modern Number Theory
2 Fundamental Concepts
- Natural Numbers
- Integers
- Rational and Irrational Numbers
- Real and Complex Numbers
3 Mathematical Proof Techniques
- Direct Proof
- Proof by Contradiction
- Mathematical Induction
- Constructive Proofs
02 Divisibility and Prime Numbers 3 topics
1 Divisibility Properties
- Division Algorithm
- Greatest Common Divisor
- Least Common Multiple
2 Prime Numbers
- Fundamental Theorem of Arithmetic
- Distribution of Primes
- Prime Number Theorem
- Sieve of Eratosthenes
3 Factorization Methods
- Trial Division
- Fermat's Factorization
- Pollard's rho Algorithm
- Quadratic Sieve
03 Congruences 3 topics
1 Modular Arithmetic
- Congruence Relations
- Linear Congruences
- Chinese Remainder Theorem
2 Euler's Theorem and Fermat's Little Theorem
- Euler's Totient Function
- Carmichael Numbers
- Pseudoprimes
3 Quadratic Residues
- Legendre Symbol
- Jacobi Symbol
- Quadratic Reciprocity Law
04 Diophantine Equations 3 topics
1 Linear Diophantine Equations
- Solutions and Methods
- Extended Euclidean Algorithm
2 Quadratic Diophantine Equations
- Pell's Equation
- Pythagorean Triples
3 Higher Degree Diophantine Equations
- Fermat's Last Theorem
- Catalan's Conjecture
- ABC Conjecture
05 Number Theoretic Functions 3 topics
1 Arithmetic Functions
- Divisor Functions
- Sum of Powers Functions
- Perfect Numbers
2 Multiplicative Functions
- Möbius Function
- Mangoldt Function
- Liouville Function
3 Additive Functions
- Properties and Examples
- Asymptotic Behavior
06 Continued Fractions 3 topics
1 Basic Theory
- Simple Continued Fractions
- Convergents and Properties
2 Applications
- Approximation of Irrational Numbers
- Periodic Continued Fractions
- Continued Fractions for Quadratic Irrationals
3 Diophantine Approximation
- Best Approximations
- Lagrange's Theorem
- Hurwitz's Theorem
07 Algebraic Number Theory 3 topics
1 Algebraic Numbers and Integers
- Minimal Polynomials
- Algebraic Extensions
- Number Fields
2 Ring of Integers
- Algebraic Integers
- Integral Basis
- Discriminants
3 Ideal Theory
- Prime Ideals
- Factorization of Ideals
- Class Group and Class Number
08 Analytic Number Theory 3 topics
1 Prime Number Theory
- Riemann Zeta Function
- Prime Number Theorem Proofs
- Explicit Formulas
2 L-Functions
- Dirichlet L-Functions
- Functional Equations
- Analytic Continuation
3 Distribution of Primes
- Gaps Between Primes
- Twin Primes Conjecture
- Goldbach Conjecture
09 Elliptic Curves 3 topics
1 Introduction to Elliptic Curves
- Weierstrass Form
- Group Structure
- Torsion Points
2 Elliptic Curves over Finite Fields
- Counting Points
- Schoof's Algorithm
- Supersingular Curves
3 Applications
- Elliptic Curve Cryptography
- Factorization Methods
- Elliptic Curve Primality Proving
10 Computational Number Theory 3 topics
1 Primality Testing
- Probabilistic Tests
- AKS Algorithm
- Proth's Theorem
2 Factorization Algorithms
- Integer Factorization Records
- Number Field Sieve
- Special Number Field Sieve
3 Algorithms for Discrete Logarithms
- Index Calculus
- Pohlig-Hellman Algorithm
- Baby-step Giant-step Algorithm
11 Advanced Topics in Number Theory 3 topics
1 Transcendental Numbers
- Liouville Numbers
- Transcendence of e and π
- Baker's Theory
2 Probabilistic Number Theory
- Normal Numbers
- Almost All Properties
- Erdős–Kac Theorem
3 Additive Number Theory
- Waring's Problem
- Goldbach Conjecture
- Partition Functions
12 Applications of Number Theory 3 topics
1 Cryptography
- RSA Algorithm
- Diffie-Hellman Key Exchange
- Post-Quantum Cryptography
2 Coding Theory
- Error-Correcting Codes
- Reed-Solomon Codes
- Number-Theoretic Codes
3 Computer Science Applications
- Hashing Functions
- Pseudo-Random Number Generation
- Quantum Computing Implications
13 Unsolved Problems in Number Theory 3 topics
1 Famous Conjectures
- Riemann Hypothesis
- Twin Prime Conjecture
- Collatz Conjecture
2 Progress on Open Problems
- Partial Results
- Computational Evidence
- Related Problems
3 Approaches and Techniques
- Analytic Methods
- Algebraic Methods
- Computational Approaches
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