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Primes are the indivisible building blocks of arithmetic, yet their distribution has puzzled mathematicians for over two thousand years. In this video we start with Euclid’s elegant proof of infinitely many primes, explore how to test whether a number is prime, and see the power of the Sieve of Eratosthenes. We follow Gauss as a 15-year-old counting primes and discovering their density is linked to the logarithm, before turning to Legendre’s correction, the logarithmic integral, and the remarkable connection π(x) ~ x/ln(x). Along the way we uncover why primes grow rarer but never run out, how compounding interest leads naturally to e and ln(x), and why these curves mirror the spacing of primes. The journey ends with Riemann’s 1859 paper and the eventual 1896 proof of the Prime Number Theorem by Hadamard and de la Vallée Poussin. This is the full story of how chaos in the primes gives way to one of the smoothest laws in mathematics.
Sources and References:
Derbyshire, J. (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN: 978-0309085496.
Dunham, W. (1990). Journey Through Genius: The Great Theorems of Mathematics. Wiley. ISBN: 978-0140147391.
Courant, R. & Robbins, H. (1941; revised editions by Ian Stewart). What is Mathematics? Oxford University Press.
Petersen, B. E. (1996). Prime Number Theorem (Seminar Lecture Notes). UC Davis.
Liu, R. Prime Number Theorem (lecture notes).
Goldstein, L. J. (1973). “A History of the Prime Number Theorem.” The American Mathematical Monthly, 80(6), 599–615
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Goldfeld, D. (1998). The Elementary Proof of the Prime Number Theorem: An Historical Perspective
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Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse (“On the Number of Primes Less Than a Given Magnitude”).
Hadamard, J. (1896). “Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques.” Bulletin de la Société Mathématique de France.
de la Vallée Poussin, C. J. (1896). “Recherches analytiques sur la théorie des nombres premiers.” Annales de la Société Scientifique de Bruxelles.
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