Number Theory
Year 3 · 4 sections · 47 concepts · 87 practice questions
What's covered
free1 · Orders and primitive roots13 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Multiplicative order
- Lemma 1.2 · Orders divide exponents
- Proposition 1.3 · The order of a power
- Lemma 1.4 · The totient sum
- Theorem 1.5 · Lagrange's theorem on polynomial congruences
- Theorem 1.6 · Primitive roots modulo a prime
- Proposition 1.7 · Testing for a primitive root
- Theorem 1.8 · Hensel's lemma
- Theorem 1.9 · Which moduli have primitive roots
- Corollary 1.10 · The structure of the unit group
- Example 1.11 · Discrete logarithms and power congruences
- Check 1.12 · Check: orders and primitive roots
2 · Quadratic reciprocity11 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · Quadratic residues and the Legendre symbol
- Theorem 2.2 · Euler's criterion
- Corollary 2.3 · When is −1 a square?
- Lemma 2.4 · Gauss's lemma
- Proposition 2.5 · When is 2 a square?
- Theorem 2.6 · The law of quadratic reciprocity
- Definition 2.7 · The Jacobi symbol
- Example 2.8 · Computing a Legendre symbol
- Corollary 2.9 · Infinitely many primes ≡ 1 mod 4
- Check 2.10 · Check: deciding whether a is a square mod p
3 · Sums of squares and Diophantine approximation11 concepts
- Overview 3 · The plan of this section
- Theorem 3.1 · Fermat's two-squares theorem
- Theorem 3.2 · Which integers are sums of two squares
- Theorem 3.3 · Lagrange's four-squares theorem
- Definition 3.4 · Continued fractions
- Proposition 3.5 · Properties of convergents
- Theorem 3.6 · Best rational approximation
- Theorem 3.7 · Periodic continued fractions
- Theorem 3.8 · Pell's equation
- Example 3.9 · Solving Pell's equation with continued fractions
- Check 3.10 · Check: quadratic Diophantine problems
4 · Arithmetic functions and the distribution of primes12 concepts
- Overview 4 · The plan of this section
- Definition 4.1 · Arithmetic and multiplicative functions
- Proposition 4.2 · Dirichlet convolution
- Lemma 4.3 · The Möbius function inverts 1
- Theorem 4.4 · Möbius inversion
- Corollary 4.5 · The totient formula
- Theorem 4.6 · Even perfect numbers (Euclid–Euler)
- Proposition 4.7 · Dirichlet series and the Euler product
- Theorem 4.8 · The prime reciprocals diverge
- Theorem 4.9 · Chebyshev's bounds
- Remark 4.10 · The prime number theorem
- Check 4.11 · Check: arithmetic functions
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Lemma 1.2 · Orders divide exponents
⇔ ; so orders divide .
Proof
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