Prime Numbers & The Sieve of Eratosthenes
An exploration of prime distribution, historical sieving algorithms, and modern computational optimizations for large-scale factorization.
Congruence Relations & Ring Structures
Dive into modular arithmetic foundations, equivalence classes, and how residue systems form algebraic structures essential for cryptography.
From Pell's Equation to Fermat's Last Theorem
A historical and mathematical journey through integer solutions, infinite descent, and Wiles' groundbreaking proof using modular forms.
The Riemann Zeta Function & Prime Density
Understanding complex analysis techniques applied to number theory, the critical line hypothesis, and implications for prime gaps.
Legendre Symbols & Quadratic Residues
Euler's criterion, Jacobi symbols, and the elegant symmetry governing solvability of quadratic congruences modulo primes.
Elliptic Curves in Modern Cryptography
How algebraic geometry meets number theory to secure digital communications, including ECC implementations and point compression.
The ABC Conjecture & Recent Advances
Examining Oishi's claimed proof, the interplay between additive and multiplicative number theory, and its impact on Fermat-type equations.
Number Fields & Rings of Integers
Generalizing ℤ to algebraic extensions, unique factorization domains, class groups, and ideal theory in quadratic fields.