Discrete Mathematics - Vol 3 | 12. Introduction to Fermat’s Little Theorem and Primality Testing by Abraham | Learn Smarter
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12. Introduction to Fermat’s Little Theorem and Primality Testing

Fermat's Little Theorem is a key result in number theory, stating that for a prime number p and an integer a not divisible by p, the expression a^(p-1) is congruent to 1 modulo p. This theorem can help in primality testing, though limitations exist, especially with certain types of composite numbers called Carmichael numbers. The chapter also delves into practical applications of the theorem in calculations and the concepts of pseudo primes and Carmichael numbers.

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Sections

  • 12.1

    Discrete Mathematics

    This section introduces Fermat's Little Theorem and its implications for primality testing and the concept of Carmichael numbers.

    Learning Practice

  • 12.1.1

    Introduction To Fermat’s Little Theorem And Primality Testing

    This section introduces Fermat’s Little Theorem and its significance in primality testing, along with a discussion on Carmichael numbers.

    Learning Practice

  • 12.1.2

    Fermat's Little Theorem

    Fermat's Little Theorem states that for a prime number p and an integer a not divisible by p, a^(p-1) is congruent to 1 modulo p.

    Learning Practice

  • 12.1.3

    Corollary Of Fermat's Little Theorem

    This section discusses Fermat's Little Theorem and its corollary, highlighting their significance in number theory and applications in primality testing.

    Learning Practice

  • 12.1.4

    Proof Of Fermat's Little Theorem

    Fermat's Little Theorem states that if a prime number p does not divide integer a, then a raised to p-1 is congruent to 1 mod p.

    Learning Practice

  • 12.1.5

    Applications Of Fermat's Little Theorem

    Fermat's Little Theorem provides a method for primality testing and has applications in modular arithmetic involving prime numbers.

    Learning Practice

  • 12.1.6

    Primality Testing Algorithms

    This section explores Fermat's Little Theorem and Carmichael numbers as foundational concepts in primality testing algorithms.

    Learning Practice

  • 12.1.7

    Carmichael Numbers

    This section explores Carmichael numbers, their properties, and their significance in number theory, particularly in relation to Fermat's Little Theorem.

    Learning Practice

  • 12.1.8

    Conclusion

    The conclusion discusses Fermat's Little Theorem, its applications in primality testing, and the significance of Carmichael numbers.

    Learning Practice

  • 12.2

    Understanding Fermat's Little Theorem

    Fermat's Little Theorem provides a way to determine properties of prime numbers and plays a crucial role in primality testing.

    Learning Practice

  • 12.2.1

    Statement And Explanation

    Fermat's Little Theorem is critical for primality testing and establishes that if a number is prime, certain mathematical properties hold.

    Learning Practice

  • 12.2.2

    Proof Overview

    This section explores Fermat's Little Theorem, its proof, and applications including primality testing and Carmichael numbers.

    Learning Practice

  • 12.3

    Applications In Primality Testing

    This section discusses Fermat's Little Theorem, its application in primality testing, and introduces the concept of Carmichael numbers.

    Learning Practice

  • 12.3.1

    Using Fermat's Little Theorem For Modular Arithmetic

    Fermat's Little Theorem provides a method for performing modular arithmetic with prime numbers, enabling efficient calculations and primality testing.

    Learning Practice

  • 12.3.2

    Primality Testing Algorithm Limitations

    This section discusses the limitations of using Fermat's Little Theorem for primality testing, introducing concepts like pseudo primes and Carmichael numbers.

    Learning Practice

  • 12.4

    Carmichael Numbers And Pseudoprimes

    This section discusses Fermat's Little Theorem and its implications for primality testing, focusing on Carmichael numbers and pseudoprimes.

    Learning Practice

  • 12.4.1

    Definition Of Pseudoprimes

    This section introduces the concept of pseudoprimes, highlighting their properties in relation to Fermat's Little Theorem.

    Learning Practice

  • 12.4.2

    Characteristics Of Carmichael Numbers

    Carmichael numbers are composite numbers that satisfy Fermat's little theorem for all bases coprime to them, making them pseudo primes.

    Learning Practice

  • 12.5

    Examples And Concluding Thoughts

    This section emphasizes Fermat's Little Theorem and its applications in primality testing, alongside an examination of Carmichael numbers.

    Learning Practice

  • 12.5.1

    Example Of A Carmichael Number (561)

    This section introduces Fermat's Little Theorem, its applications in primality testing, and discusses the properties and significance of Carmichael numbers, particularly highlighting the number 561 as an example.

    Learning Practice

  • 12.5.2

    Final Remarks On Number Theory

    This section elaborates on Fermat's Little Theorem, its implications for primality testing, and discusses Carmichael numbers which challenge these tests.

    Learning Practice

References

ch61.pdf

Class Notes

Memorization

What we have learnt

  • Fermat's Little Theorem pro...
  • The theorem can be used to ...
  • Carmichael numbers can caus...

Final Test

Revision Tests