Primitive Roots Modulo n

Learn the fundamental concepts of primitive roots in modular arithmetic through this comprehensive number theory lecture series. Explore the definition and properties of primitive roots modulo n, starting with the order of integers modulo n and progressing through key propositions and examples. Master techniques for finding primitive roots modulo various numbers including primes, prime powers, and composite numbers like 22 and 54. Discover when primitive roots exist modulo n and understand the critical theorem that there are no primitive roots modulo mn when gcd(m,n)=1. Examine the existence of primitive roots modulo powers of odd primes and modulo 2p^n, while working through technical lemmas that support these results. Apply primitive root theory to solve exponential congruences and explore the concept of discrete logarithms through the index function. Investigate the connection between primitive roots and the multiplicative group structure U(n), including solutions to polynomial congruences of the form x^d ≡ 1 (mod p) and the existence of mth roots modulo n.

Syllabus

Number Theory | Order of an integer modulo n: Proposition 1
Number Theory | Primitive Roots modulo n: Definition and Examples
Number Theory | Order of an element modulo n Proposition 2
Number Theory | Order of a^k mod n
Number Theory | Number of Primitive Roots modulo n
Number Theory | All primitive roots modulo 22
Number Theory | Solutions of x^d-1 =0 mod p.
Number Theory | Integers of order d mod p
Number Theory | Products of primitive roots modulo p
Number Theory | If gcd(m,n)=1 there are no primitive roots modulo mn!!
Number Theory | Primitive roots modulo p -- Technical Lemma 1
Number Theory | Primitive roots modulo p -- Technical Lemma 2
Number Theory | There is a primitive root modulo every power of an odd prime!!
Number Theory | There is a primitive root modulo 2p^n!!
Number Theory | When are there primitive roots modulo n?
Number Theory | Primitive roots of 54
Number Theory | A primitive root for 2*5^n
Number Theory | Definition of the index with respect to a primitive root modulo n.
Number Theory | The discrete root -- Example 2
Number Theory | Existence of an mth root modulo n
Number Theory | Solving an exponential congruence with primitive roots: Example 1
Number Theory | Solving an exponential congruence using primitive roots: Example 2.
Number Theory | Existence of a solution to an exponential congruence-- Example
Primitive roots modulo n and the structure of U(n)