Why do prime numbers make these spirals? | Dirichlet's theorem and pi approximations
by Grant Sanderson · 3Blue1Brown
Beginner entry point that reaches a real theorem on prime distribution, not trivia. A 22-minute visual explanation of why primes plotted in polar coordinates form spirals and rays. It moves from rational approximations of 2π to residue classes, Euler's totient function, and Dirichlet's theorem on primes in arithmetic progressions.
More resources on Prime Numbers
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
primes.utm.edu
Primes.utm.edu, The Prime Pages, is a comprehensive number theory resource focused on primes, offering extensive prime catalogs (including Mersenne primes), historical notes, tables, and references to algorithms, software, and research data.
OEIS Prime Sequences
Search results for prime-related entries in the On-Line Encyclopedia of Integer Sequences, the standard database for number sequences. Each entry gives terms, formulas, references and code, letting readers identify a sequence met in their own work and trace its literature.
But what is the Riemann zeta function? Visualizing analytic continuation
Animated 3Blue1Brown explanation of how the zeta function is defined for complex inputs and extended beyond its region of convergence by analytic continuation. Viewers come to understand what the Riemann hypothesis actually claims and why it connects to the primes.
Prime Numbers: A Computational Perspective (2nd Edition)
A graduate-level reference (Springer, 2005) on computing with primes. It covers prime distribution, primality proving including AKS, factoring algorithms such as the quadratic and number field sieves, elliptic-curve methods, and fast arithmetic, with over 100 algorithms in pseudocode.
The distribution of prime numbers
Short, rigorous, free survey of prime distribution. Expository lecture notes of about 20 pages from a NATO summer school in Montreal. They cover the prime number theorem, Cramér's probabilistic model and where it breaks down, primes in short intervals, and how primes are distributed in arithmetic progressions.