The Yellowstone Permutation - Numberphile

TL;DR

The Yellowstone Permutation is a sequence with specific rules where every number eventually appears, including "geysers" that erupt periodically.

Transcript

Okay so you just got back from Iceland? Where  you saw some geysers. So I am going to tell you   about a sequence that has a lot of geysers in it,  it's called the Yellowstone permutation. Why it's   called a permutation will emerge in a minute. And  in this one you're actually told the first three   terms and then the rules take over. Rules are  v... Read More

Key Insights

• 🍉 The Yellowstone Permutation is a sequence with simple rules that determine each term based on the previous terms.
• ⌛ Geysers, which are numbers three times a prime, periodically erupt and appear as spikes in the sequence.
• #️⃣ The sequence is infinite, and every number eventually appears, including geysers.
• ❓ The appearance of geysers is related to the presence of primes in the sequence.
• 🦕 Most of the terms in the sequence follow an alternating pattern of even and odd numbers, but geysers disrupt this pattern.
• 😚 The sequence can be visualized on a graph, with primes appearing close to a line with a slope of 1/2 and other numbers appearing close to the diagonal line.
• 👍 There is a theory that explains the pattern and disruptions in the sequence, but it has yet to be formally proven.

Questions & Answers

Q: How does the Yellowstone Permutation sequence work?

The sequence starts with the numbers 1, 2, and 3. Each subsequent term is determined by being relatively prime to the previous term and having a common factor with the term two steps back.

Q: Why are some numbers referred to as "geysers"?

Geysers are numbers in the sequence that are three times a prime. They periodically erupt and appear as spikes in the sequence.

Q: Does every number eventually appear in the Yellowstone Permutation sequence?

Yes, every number eventually appears in the sequence. This is proven by showing that there are infinitely many primes in the sequence and that each prime divides infinitely many terms.

Q: How do geysers affect the sequence?

Geysers, which are numbers three times a prime, create disruptions in the otherwise alternating pattern of even and odd numbers in the sequence. They appear periodically and can be arbitrarily large.

Summary & Key Takeaways

• The Yellowstone Permutation is a sequence of numbers where each term must be relatively prime to the previous term and have a common factor with the term two steps back.

• The sequence starts with 1, 2, and 3, and the rules are applied to determine subsequent terms.

• Geysers, which are numbers three times a prime, periodically erupt in the sequence.