Primes and Hidden Order — From the Sieve to Bounded Gaps
Primes resist one-step prediction, yet they are not featureless randomness. Follow proofs of infinitude and the sieve in Alexandria, congruences in Toulouse, products and series in Saint Petersburg, arithmetic progressions and zeta zeros in Berlin, average density in Paris and Brussels, and modern sieve and gap problems in Oslo, Princeton, Beijing, Cambridge, and Durham. The route keeps local pattern, average law, infinite theorem, and open conjecture as different claims.
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c. 300 BCE
Alexandria
Proving That the Primes Never End — Book IX of the Elements
Proposition 20 of Euclid’s Book IX shows that however many primes are given, those alone cannot account for every number. It is often summarized as ‘multiply them all and add one,’ but the original argument uses a prime divisor of a number one greater than the product. Infinitely many primes is neither a formula for the next prime nor the whole modern theorem of unique factorization.
Replay on the map from scene 1Scene index