Number Theory is the study of the properties and relationships of integers — particularly prime numbers. Often called the “queen of mathematics.”

Key Topics

• Divisibility and Primes — The fundamental theorem of arithmetic: every integer greater than 1 is a unique product of primes.
• Modular Arithmetic — Arithmetic on remainders: \(a \equiv b \pmod{n}\).
• Diophantine Equations — Finding integer solutions to equations like \(ax + by = c\).
• Euler's Totient Function — Counting integers coprime to \(n\): \(\varphi(n)\).
• Fermat's Little Theorem and Euler's Theorem — Foundational results for cryptography.
• Quadratic Reciprocity — A deep theorem about when equations \(x^2 \equiv a \pmod{p}\) have solutions.
• Continued Fractions — Representing numbers as nested fractions, best rational approximations.
• The Riemann Zeta Function — \(\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\) and the distribution of primes.

Why It Matters

Number theory is the foundation of modern cryptography (RSA, Diffie-Hellman, elliptic curve cryptography). Every secure internet transaction relies on the difficulty of certain number-theoretic problems.