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Number Theory
Explore divisibility and congruences; arithmetic functions and equations; prime distribution and advanced connections. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Fields
A field is a commutative ring where every nonzero element has a multiplicative inverse.
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Divisibility and congruences
Prime factorization
Every integer greater than one factors uniquely into primes up to their order.
Video explanations to be addedGreatest common divisor
For integers a and b that are not both zero, the greatest common divisor is the unique positive integer dividing both that is divisible by every other common divisor. Signs do not change its positive value.
8 reviewed resourcesEuclidean algorithm
The Euclidean algorithm computes a greatest common divisor by repeated division with remainder. Replacing (a,b) by (b,r) preserves the common divisors, and the strictly smaller nonnegative remainders force termination.
6 reviewed resourcesModular arithmetic
Congruence identifies integers with the same remainder modulo a positive integer.
Video explanations to be addedModular inverses
An integer has an inverse modulo n exactly when it is coprime to n.
Video explanations to be addedChinese remainder theorem
Pairwise coprime moduli give a unique combined congruence class modulo their product.
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Arithmetic functions and equations
Euler's totient
Euler's totient counts positive residues coprime to a positive integer modulus.
Video explanations to be addedFermat and Euler theorems
Coprime residues satisfy power congruences controlled by prime moduli or Euler's totient.
Video explanations to be addedDiophantine equations
Diophantine equations seek integer or rational solutions rather than unrestricted real solutions.
Video explanations to be addedQuadratic residues
A residue is quadratic when it is congruent to a square modulo the chosen integer.
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Prime distribution and advanced connections
Prime number theorem
The number of primes up to x is asymptotic to x divided by log x.
Video explanations to be addedRiemann zeta function
A Dirichlet series over positive integers defines zeta in a convergent half-plane and admits analytic continuation.
Video explanations to be addedAlgebraic numbers
An algebraic number is a root of a nonzero polynomial with rational coefficients.
Video explanations to be addedFinite fields
A finite field has prime-power size; fields of each such size exist and are unique up to isomorphism.
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