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20 GUIDED INVESTIGATIONS

Number Theory Labs

Work through the course in order, or choose a topic that interests you. Each lab combines mathematical questions with programs from the Apps collection.

01๐Ÿงฎ

GCDs and the Euclidean Algorithm

Greatest common divisors, linear combinations, and the Euclidean algorithm.โ†’02๐Ÿงฑ

Factorization and Prime Numbers

Prime factorizations, the sieve of Eratosthenes, and patterns among primes.โ†’03๐Ÿ”ข

Congruences

Arithmetic with residue classes and complete residue systems.โ†’04โž•

Sums of Two Squares

Which integers can be written as a sum of two squares?โ†’05๐Ÿ“

Solutions of Congruences and Binomial Coefficients

Polynomial congruences, Pascalโ€™s triangle, and modular patterns.โ†’06๐Ÿงฉ

Linear Congruences and the Chinese Remainder Theorem

Solve one congruence or combine several compatible ones.โ†’07๐Ÿ”

Powering Algorithms and Primality Testing

Fast modular powers, pseudoprimes, and practical primality tests.โ†’08๐Ÿงฐ

Factoring Strategies

Compare Fermat, Pollard rho, and p โˆ’ 1 factorization methods.โ†’09๐Ÿ”

RSA Public Key Cryptography

Encrypt and decrypt messages using modular arithmetic.โ†’10๐Ÿชœ

Henselโ€™s Lemma

Lift solutions of congruences from p to higher prime powers.โ†’11๐Ÿ”„

Power Residues and Primitive Roots

Multiplicative orders, power tables, and primitive roots.โ†’12๐Ÿ“

Indices โ€” The Discrete Logarithm

Index tables and algorithms for the discrete logarithm problem.โ†’13โœ…

Proving Primality

Turn probable-prime evidence into a proof of primality.โ†’14โˆš

Square Roots Modulo p

Find modular square roots and examine when they exist.โ†’15โ–ฆ๏ธ

Quadratic Residues

Jacobi symbols, quadratic reciprocity, and residue patterns.โ†’16๐Ÿ“

Binary Quadratic Forms

Reduction, equivalence, and classes of binary quadratic forms.โ†’17๐Ÿ“Š

Arithmetic Functions

Explore ฯ†, ฮผ, ฯƒ, ฯ„, and other functions of an integer.โ†’18๐”ผ

Quadratic Extension Rings

Arithmetic, units, and factorization in quadratic rings.โ†’19โ€–ยทโ€–

Norms in Quadratic Extension Rings

Use norms to study elements and factorizations in quadratic rings.โ†’20๐Ÿชœ

Continued Fractions

Expansions, convergents, approximation, and Pell equations.โ†’