Escalera de Matemáticas · Universidad

Teoría de Números

Divisibilidad y el algoritmo de Euclides, primos, aritmética modular, los teoremas de Fermat y Euler, RSA, residuos cuadráticos y fracciones continuas.

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41 temas en 8 unidades. Se apoya en Matemática Discreta. Lleva a Álgebra Abstracta. ¿No estás seguro de que sea el curso adecuado? Encuentra tu peldaño.

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Unidad 1 Divisibility and the Euclidean Algorithm

6 temas

Divide with remainders, find greatest common divisors quickly, and solve ax + by = c in whole numbers.

  1. Divisibility
  2. The Division Algorithm
  3. Writing Numbers in Other Bases
  4. The Euclidean Algorithm
  5. Bézout's Identity and the Extended Euclidean Algorithm
  6. Linear Diophantine Equations

Unidad 2 Primes and Factorization

8 temas

Test for primes, factor uniquely, and use factorizations to count divisors and study how primes are spread out.

  1. Testing Primes and the Sieve of Eratosthenes
  2. Unique Prime Factorization
  3. gcd and lcm from Prime Factorizations
  4. Counting and Adding Divisors
  5. Perfect Numbers and Mersenne Primes
  6. Prime Powers in n!
  7. Infinitely Many Primes
  8. How Common Are Primes?

Unidad 3 Congruences

7 temas

Calculate with remainders: modular arithmetic, fast powers, inverses, linear congruences and the Chinese Remainder Theorem.

  1. Congruence mod n
  2. Arithmetic mod n
  3. Divisibility Tests from Congruences
  4. Fast Powers mod n
  5. Modular Inverses
  6. Linear Congruences
  7. The Chinese Remainder Theorem

Unidad 4 Fermat, Euler and Wilson

5 temas

Tame huge powers with the theorems of Fermat and Euler, count units with φ, and see why passing Fermat's test does not prove a number prime.

  1. Fermat's Little Theorem
  2. Wilson's Theorem
  3. Euler's Phi Function
  4. Euler's Theorem
  5. Fermat Tests and Pseudoprimes

Unidad 5 Orders and Primitive Roots

3 temas

Find how long the powers of a number take to repeat, find primitive roots, and meet discrete logarithms.

  1. The Order of an Element
  2. Primitive Roots
  3. Discrete Logarithms

Unidad 6 Cryptography

3 temas

Put modular arithmetic to work in ciphers, key exchange and RSA.

  1. Shift and Affine Ciphers
  2. Diffie–Hellman Key Exchange
  3. The RSA Cryptosystem

Unidad 7 Quadratic Residues

4 temas

Decide which numbers are perfect squares mod p, with Euler's criterion and the law of quadratic reciprocity.

  1. Squares mod p
  2. The Legendre Symbol and Euler's Criterion
  3. Is −1 or 2 a Square mod p?
  4. Quadratic Reciprocity

Unidad 8 Diophantine Equations and Continued Fractions

5 temas

Find whole-number solutions of x² + y² = z², sums of two squares and Pell's equation, with continued fractions as the tool.

  1. Pythagorean Triples
  2. Sums of Two Squares
  3. Continued Fractions of Rational Numbers
  4. Continued Fractions of Square Roots
  5. Pell's Equation