Mathematical Theorems
Hahn-Banach Theorem
Separates convex sets. Foundation of optimization.
Chinese Remainder Theorem
Ancient algorithm. Modern cryptography foundation.
Cauchy-Schwarz Inequality
Fundamental inequality. Defines angle in inner product spaces.
Bolzano-Weierstrass Theorem
Foundation of analysis. Compactness in real numbers.
Riemann Hypothesis
Unproven. Predicts prime distribution precisely. Million dollar prize.
Poincaré Conjecture
Perelman proved using Ricci flow. Millennium Prize problem.
Banach-Tarski Paradox
Uses axiom of choice. Demonstrates measure theory limits.
Fundamental Theorem of Algebra
Guarantees polynomial equations solvable in complex numbers.
Euler's Identity
Most beautiful equation. Connects five fundamental constants.
Four Color Theorem
Any planar graph 4-colorable. First major computer-assisted proof.
Cantor's Theorem
Infinite sets have different sizes. Revolutionized understanding of infinity.
Prime Number Theorem
Describes distribution of prime numbers. Foundation of number theory.
Gödel's Incompleteness Theorems
No system can prove all truths about arithmetic. Limits of formal systems.
Fermat's Last Theorem
Proved by Andrew Wiles after 358 years. Deepened understanding of elliptic curves.