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Videos for my online elementary number theory course.

Prime Factorization for Natural Numbers

Prime Factorization for Natural Numbers

Greatest Common Divisors (Part 1)

Greatest Common Divisors (Part 1)

Euclid’s Lemma (GCD Part 2)

Euclid’s Lemma (GCD Part 2)

Factoring into Products of Primes, and Quick Notes About RSA Challenges

Factoring into Products of Primes, and Quick Notes About RSA Challenges

Proof of the Fundamental Theorem of Arithmetic

Proof of the Fundamental Theorem of Arithmetic

Proof that there are infinitely many primes

Proof that there are infinitely many primes

Prime Number Sieve and the Definition of a Mersenne Prime

Prime Number Sieve and the Definition of a Mersenne Prime

Primes of the form ax+b

Primes of the form ax+b

A few words on the prime number theorem and the Riemann hypothesis

A few words on the prime number theorem and the Riemann hypothesis

Groups, rings, and congruence mod n

Groups, rings, and congruence mod n

Linear equations mod n

Linear equations mod n

The group of units mod n, and the order of an element in a group

The group of units mod n, and the order of an element in a group

Euler’s phi function and Euler’s Theorem

Euler’s phi function and Euler’s Theorem

Wilson’s Theorem

Wilson’s Theorem

The Chinese Remainder Theorem

The Chinese Remainder Theorem

Multiplicative functions and Euler’s phi function revisited

Multiplicative functions and Euler’s phi function revisited

GCD(a,b) as a linear combination of a and b, and computing inverse Modulo n

GCD(a,b) as a linear combination of a and b, and computing inverse Modulo n

Writing a number in binary, by hand and using Sage

Writing a number in binary, by hand and using Sage

Computing Huge Powers Mod n

Computing Huge Powers Mod n

Pseudoprimality

Pseudoprimality

Miller-Rabin and Lucas-Lehmer Primality Tests

Miller-Rabin and Lucas-Lehmer Primality Tests

Intro to cyclic groups and the group of units mod p

Intro to cyclic groups and the group of units mod p

Polynomials over Z/pZ

Polynomials over Z/pZ

The Existence of Primitive Roots mod p

The Existence of Primitive Roots mod p

Recap of Primitive Elements in (Z/nZ)^* with examples in Sage

Recap of Primitive Elements in (Z/nZ)^* with examples in Sage

Intro to Public Key Cryptography with an example

Intro to Public Key Cryptography with an example

The Diffie-Hellman Key Exchange

The Diffie-Hellman Key Exchange

More Diffie-Hellman, the Man in the Middle Attack, and finding primitive roots for certain primes p

More Diffie-Hellman, the Man in the Middle Attack, and finding primitive roots for certain primes p

Intro to RSA with an Example

Intro to RSA with an Example

Encoding and Decoding Phrases and Numbers

Encoding and Decoding Phrases and Numbers

Encoding, RSA encryption/decryption, Decoding Example

Encoding, RSA encryption/decryption, Decoding Example

Attacking RSA by Factoring n Given phi(n)

Attacking RSA by Factoring n Given phi(n)

Attacking RSA when p and q are close

Attacking RSA when p and q are close

Intro to Quadratic Reciprocity

Intro to Quadratic Reciprocity

Intro to the Legendre Symbol

Intro to the Legendre Symbol

The Legendre Symbol and Quadratic Reciprocity with Examples

The Legendre Symbol and Quadratic Reciprocity with Examples

Euler’s Criterion

Euler’s Criterion

Gauss’s Lemma

Gauss’s Lemma

Euler’s Proposition

Euler’s Proposition

Jupyter notebook example

Jupyter notebook example

Intro to Continued Fractions

Intro to Continued Fractions

Euler’s phi(pq) for primes p and q

Euler’s phi(pq) for primes p and q

Partial Convergents

Partial Convergents

The Sequence of Partial Convergents

The Sequence of Partial Convergents

Every Nonzero Rational Number Has A Simple Continued Fraction Representation

Every Nonzero Rational Number Has A Simple Continued Fraction Representation

The Continued Fraction Procedure

The Continued Fraction Procedure

Quadratic Irrationals and Period Continued Fractions

Quadratic Irrationals and Period Continued Fractions

Intro to Elliptic Curves

Intro to Elliptic Curves

The Elliptic Curve Group Law (with examples)

The Elliptic Curve Group Law (with examples)

Pollard’s p-1 Method for Factoring Integers

Pollard’s p-1 Method for Factoring Integers

The Elliptic Curve Method to Factor Integers

The Elliptic Curve Method to Factor Integers

ElGamal Encryption and Elliptic Curve Cryptography

ElGamal Encryption and Elliptic Curve Cryptography