4211 - The Party Pooper Prime - Numberphile

TL;DR

A new game covering prime-indexed points with the fewest straight lines reveals 'awkward primes' that force a new line, culminating in the 'party pooper prime' 4211.

Key Points

• 1.Prime numbers are simultaneously irregular and surprisingly smooth. The count of primes up to n, written π(n), follows n/log(n) so closely that plotting it looks nearly linear — which mathematicians find astonishing given how chaotically primes are distributed locally.
• 2.The 'line game' plots primes as points (k, p_k) and asks how few straight lines cover the first n prime-points. This is an instance of the NP-complete Set Cover problem, tracked as OEIS sequence A373813, and was computed by Max Alex up to the 861st prime.
• 3.Long flat stretches in the sequence occur when a single 'golden line' passes through many consecutive prime-points. A streak of 48 consecutive primes needed only 68 lines, and a whopping 112 primes in a row were covered by just 69 lines — a remarkable sweet spot.
• 4.'Awkward primes' are primes that force a step up in the number of lines required. Coined on camera during filming, they now have their own OEIS entry, making the host a 'co-father' of the term.
• 5.The 'party pooper prime' 4211 is the most awkward prime of all — it ends the record streak of 112 primes covered by 69 lines. It and the awkward primes collectively are listed in the Online Encyclopedia of Integer Sequences, giving 4211 its memorable title.