The High Schooler Who Solved a Prime Number Theorem

Title: The High Schooler Who Solved a Prime Number Theorem

Step-by-step Tutorial:

1. Introduction to Carmichael Numbers:

   • A Carmichael number is a number "n" such that for all integers "a", a to the power of n minus a is divisible by n.
   • Additionally, a Carmichael number is not a prime number.
   • Carmichael numbers are constructed from prime factors that have a special property.

2. Understanding the Problem:

   • The speaker mentions that math is about finding connections between seemingly unrelated things.
   • The video discusses the speaker's interest in Carmichael numbers during high school.

3. Exploring Prime Number Theory:

   • The speaker mentions the existence of a postulate for primes that states for every large number, there exists a double.
   • The speaker aims to apply a similar concept to Carmichael numbers.

4. Utilizing Mathematical Breakthroughs:

   • The speaker references a breakthrough by John Maynard related to small gaps between primes.
   • By utilizing techniques from number theory and gaps between primes, the speaker was able to make progress in studying Carmichael numbers.

5. Proof Development:

   • The speaker realized the potential of using results about gaps between primes to study Carmichael numbers.
   • They showed that for sufficiently large "n", there exist Carmichael numbers between "n" and 2n.

6. Seeking Expert Feedback:

   • Upon encountering a mistake, the speaker sent their paper to Granville, a leading expert in the field.
   • The feedback received from Daniel Lawson was technical but provided valuable insights and ideas for improvement.

7. Incorporating Fourier Analysis:

   • The speaker's proof heavily relies on Fourier analysis, a theory from physics that involves decomposing functions into waves.
   • Fourier analysis enabled the speaker to make significant claims and predictions about Carmichael numbers based on the properties of prime numbers.

8. Relevance to Cryptography:

   • Carmichael numbers have implications for cryptography, where large primes are used for encryption.
   • Understanding Carmichael numbers better can help in avoiding weaker encryption methods that may be vulnerable to attacks.

9. Community Support and Encouragement:

   • The speaker highlights the supportive nature of the number theory community, where established experts like Maynard and Granville are open to engaging with newcomers.
   • Encourages individuals to explore mathematical problems and reach out to experts in the field for guidance and collaboration.

10. Conclusion:

• The video concludes with the speaker expressing gratitude for the support received from renowned mathematicians and encourages others to engage in mathematical problem-solving and reach out for guidance when needed.