How to Learn Number Theory
A structured path through Number Theory — from first principles to confident mastery. Check off each milestone as you go.
Number Theory Learning Roadmap
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Foundations: Divisibility and the Integers
Study the division algorithm, divisibility, greatest common divisors, the Euclidean Algorithm, and the Fundamental Theorem of Arithmetic. Build comfort with mathematical proof techniques including induction.
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Modular Arithmetic and Congruences
Master congruence notation, modular operations, linear congruences, and the Chinese Remainder Theorem. Practice solving systems of congruences and understand residue classes.
Key Theorems: Fermat, Euler, and Wilson
Learn Fermat's Little Theorem, Euler's Theorem, Wilson's Theorem, and their applications. Understand Euler's totient function and its properties for prime powers and products.
Quadratic Residues and Reciprocity
Study the Legendre symbol, Euler's criterion, and the Law of Quadratic Reciprocity. Explore quadratic residues modulo primes and composite numbers.
Arithmetic Functions and Multiplicativity
Investigate number-theoretic functions such as the divisor function, the sum-of-divisors function, the Mobius function, and Dirichlet convolution. Understand Mobius inversion.
Prime Distribution and Analytic Methods
Learn the Prime Number Theorem, Chebyshev's estimates, and the role of the Riemann zeta function. Get an introduction to Dirichlet's Theorem on primes in arithmetic progressions.
Cryptographic Applications
Study RSA encryption, the Diffie-Hellman key exchange, elliptic-curve cryptography, and primality testing algorithms (Miller-Rabin, AKS). Understand computational complexity in number theory.
Advanced Topics and Open Problems
Explore algebraic number theory (rings of integers, ideals, class numbers), p-adic numbers, elliptic curves over the rationals, and major open problems including the Riemann Hypothesis and Goldbach's Conjecture.
Explore your way
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one: