The Goldbach Conjecture: State of the Art
An in-depth exploration of one of mathematics' oldest unsolved problems, including Chen's theorem, modern sieve methods, and computational verification up to 4×10¹⁸.
The branch of mathematics devoted to the study of integers and integer-valued functions. Often called the "queen of mathematics," number theory explores prime numbers, Diophantine equations, modular arithmetic, and the deep structural properties of counting numbers.
An in-depth exploration of one of mathematics' oldest unsolved problems, including Chen's theorem, modern sieve methods, and computational verification up to 4×10¹⁸.
How Legendre, Gauss, and Hadamard unlocked the density of primes. Covers the logarithmic integral, error terms, and connections to complex analysis.
Solving x² - Dy² = 1 through convergents of √D. Historical context from Bhāskara II to Lagrange, with algorithmic approaches and Python implementations.
The bridge between number theory and geometry. Explores the Taniyama-Shimura conjecture, elliptic curves, and Wiles' groundbreaking proof strategy.
Euler's discovery, Gauss's six proofs, and why it remains a cornerstone of elementary number theory. Includes visual proofs and modern generalizations.
From Euler's product formula to the analytic continuation. Examines the critical strip, trivial vs non-trivial zeros, and implications for prime distribution.