Discrete Mathematics - Vol 3 | 8. Prime Numbers and GCD by Abraham | Learn Smarter
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8. Prime Numbers and GCD

The chapter discusses prime numbers, their properties, and a naive algorithm for primality testing. It also introduces the concept of the greatest common divisor (GCD) and details Euclid's GCD algorithm, highlighting its polynomial time complexity in relation to the number of bits required to represent integers. Key algorithms and their efficiencies are compared and explained.

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Sections

  • 8

    Prime Numbers And Gcd

    This section introduces prime numbers, details the naive algorithm for primality testing, and explains the concept of the greatest common divisor (GCD) along with Euclid's GCD algorithm.

    Learning Practice

  • 8.1

    Introduction To Prime Numbers

    This section introduces prime numbers, their properties, the naive algorithm for primality testing, and the concept of Greatest Common Divisor (GCD) along with Euclid's GCD algorithm.

    Learning Practice

  • 8.2

    Properties Of Prime Numbers

    This section provides an overview of prime numbers, their properties, naive primality testing, and the Greatest Common Divisor (GCD).

    Learning Practice

  • 1.3

    Primality Testing

    This section introduces prime numbers and discusses primality testing algorithms, including a naive method and an advanced polynomial time algorithm.

    Learning Practice

  • 8.4

    Naive Algorithm For Primality Testing

    This section discusses prime numbers, properties of primes, and the naive algorithm for primality testing, highlighting its complexity.

    Learning Practice

  • 8.5

    Running Time Of The Naive Algorithm

    This section discusses the naive algorithm for primality testing, highlighting its running time and comparing it to more efficient algorithms.

    Learning Practice

  • 8.6

    Aks Primality Testing Algorithm

    This section discusses the AKS primality testing algorithm, detailing its polynomial time efficiency compared to naive algorithms for determining primality.

    Learning Practice

  • 8.7

    Greatest Common Divisor (Gcd)

    This section covers the concept of the Greatest Common Divisor (GCD), its mathematical significance, and the efficient algorithm (Euclid's algorithm) to compute it.

    Learning Practice

  • 8.7.1

    Definition Of Gcd

    The GCD (Greatest Common Divisor) is defined as the largest integer that divides two given integers without leaving a remainder.

    Learning Practice

  • 8.7.2

    Finding Gcd Using Prime Factorization

    This section discusses the concepts of prime numbers and the methods for finding the greatest common divisor (GCD) using both prime factorization and Euclid's algorithm.

    Learning Practice

  • 8.7.3

    Euclid’s Gcd Algorithm

    Euclid's GCD algorithm is a fundamental method for computing the greatest common divisor (GCD) of two integers by leveraging the properties of divisors and remainders.

    Learning Practice

  • 8.7.4

    Properties Of Gcd

    This section discusses the definition of the greatest common divisor (GCD), its properties, and Euclid's algorithm, highlighting its significance in computing GCD efficiently.

    Learning Practice

  • 8.7.5

    Proof Of Euclid’s Gcd Algorithm

    This section introduces Euclid’s GCD algorithm, explaining its properties, correctness, and polynomial time complexity compared to naive algorithms.

    Learning Practice

  • 8.7.6

    Running Time Of The Euclid Gcd Algorithm

    This section discusses the Euclid GCD algorithm, its efficiency, and its significance in computational mathematics.

    Learning Practice

References

ch57.pdf

Class Notes

Memorization

What we have learnt

  • Prime numbers are defined a...
  • The naive algorithm for pri...
  • Euclid's GCD algorithm is a...

Final Test

Revision Tests