Paterson Primes (with 3Blue1Brown) - Numberphile

TL;DR

Using a base 4 method to generate prime numbers can yield interesting results but is not foolproof.

Transcript

So this one I want to go back to when I was  a student, I was in high school, and a friend   of mine who knew I was kind of a math  person came up and he's like, hey Grant   do you get like money if you can find really big  prime numbers? Like I don't know maybe. He's like,   because I think I found a way that you can get uh  uh any any arbitrarily... Read More

Key Insights

• 👶 The base 4 prime number generation method shows promise initially but fails for larger primes due to the emergence of new divisors.
• ⚾ The method relies on unique properties of base 10 and base 4 in generating and testing prime numbers.
• 🖐️ Divisibility rules play a significant role in determining the accuracy and reliability of the generated prime numbers.
• 🌥️ Limitations of the Paterson method become apparent when larger prime numbers are analyzed.
• #️⃣ The method highlights the intricate relationship between number bases and prime number generation.
• #️⃣ Small numbers exhibit patterns that may not hold true for larger numbers, leading to the breakdown of the Paterson method.
• 📏 Understanding the underlying rules of divisibility is essential for evaluating the effectiveness of prime number generation methods.

Questions & Answers

Q: How does the base 4 method for generating prime numbers work?

The method involves converting prime numbers to base 4, where patterns are observed to generate new prime numbers in base 10.

Q: Why does the base 4 method eventually break down for larger prime numbers?

The method fails to guarantee primality for larger primes as divisors above 5 can emerge, making the method unreliable.

Q: What is the significance of the divisibility rules for base 10 and base 4 in this context?

Divisibility rules play a crucial role, as numbers not divisible by 2, 3, or 5 in base 10 remain unaffected in base 4, affecting the generated prime numbers' composition.

Q: What is the key limitation of the Paterson prime number generation method?

The method's main limitation lies in not being able to ensure the absence of prime factors like 2, 3, or 5 in the generated prime numbers, making it unreliable for larger numbers.

Summary & Key Takeaways

• A method using base 4 to generate prime numbers was introduced by Patrick Paterson.

• The method involves converting prime numbers to base 4 and checking for primality in base 10.

• While the method initially seemed to work, it eventually breaks down for larger numbers.