4211 - The Party Pooper Prime - Numberphile

This discussion introduces a unique perspective on prime numbers, exploring their irregular yet surprisingly regular distribution. While primes appear haphazardly on the number line, growing "like weeds" with unpredictable gaps, their overall distribution, when viewed on a large scale (e.g., between 1 and a million), exhibits remarkable smoothness. This global behavior is captured by the prime-counting function, pi(n), which approximates n/log n. The linearity of this function is considered astonishing, given the initial irregularity of primes. The core idea presented is to visualize primes as points on a graph, where the x-coordinate is the index 'k' and the y-coordinate is the k-th prime number (prime_k). For instance, the first prime (2) is plotted at (1, 2), the second prime (3) at (2, 3), and so on. The objective is to determine how many straight lines are needed to cover these prime points, using the minimum number of lines possible for the first 'n' primes. Let's illustrate with examples: - For one prime (n=1), one line is needed. - For two primes (n=2), a single line can connect (1, 2) and (2, 3). - For three primes (n=3), a line can connect (1, 2) and (2, 3), but the third prime (5) at (3, 5) is not on this line. Similarly, a line connecting (2, 3) and (3, 5) does not include (1, 2). Therefore, two lines are required. - For four primes (n=4), the points are (1, 2), (2, 3), (3, 5), (4, 7). Interestingly, the primes 3, 5, and 7 are collinear, meaning a single line can cover (2, 3), (3, 5), and (4, 7). This allows covering all four primes with two lines: one for (2, 3), (3, 5), (4, 7) and another for (1, 2). - For five primes (n=5), the points are (1, 2), (2, 3), (3, 5), (4, 7), (5, 11). Again, two lines suffice: one for (2, 3), (3, 5), (4, 7) and another for (1, 2) and (5, 11). It's important to note that these lines don't have to be connected; they are independent line segments. The problem of covering these prime points with the fewest lines is analogous to the famous computer science problem known as "set cover," which is an NP-complete problem. This means it's computationally challenging to find the optimal solution, but efficient algorithms exist. A computer program was used to calculate the number of lines needed for the first 410 primes. The results show an increasing, somewhat irregular sequence with "long flat stretches." These flat stretches occur when a single line covers multiple primes, thereby delaying the need for an additional line. The sequence has been calculated up to the 861st prime, revealing two significant "golden lines": one covering 48 consecutive primes using 68 lines, and another covering 112 primes using 69 lines. Beyond these, no other such dense lines have been found. The sequence of the number of lines appears roughly linear, possibly increasing at a rate of x/log x, which is consistent with the nature of prime numbers. The moments where the number of lines increases are caused by "awkward primes" – primes that cannot be covered by existing lines and thus necessitate a new line. This concept of "awkward primes" was coined during the discussion and has since been added to the Online Encyclopedia of Integer Sequences (OEIS) as sequence A373813, with a specific "party pooper prime" highlighted as the one ending a long streak of 112 primes covered by a single line. The discussion concludes with a "puzzle alert" from Jane Street, the channel's sponsor, challenging viewers to apply their number skills to a related problem. Jane Street is a quantitative trading firm that values problem-solving abilities over specific financial backgrounds, offering various career opportunities. The video also briefly mentions other mathematical sequences, such as the Van Eck sequence, as examples of similar mathematical explorations.