Proofs of Existence and Optimality · Difficult Problems in Number Theory (Optional)

Lesson 2

Nikolai Chukhin · Alexander S. Kulikov

Problem. Pierre de Fermat conjectured that, for any \(n \in \mathbb{Z}_{\ge 0}\), the number \(2^{2^n}+1\) is prime, and Leonhard Euler found the smallest \(n\) for which this is not the case (in times when there were no computers!). Find this value of \(n\).

For the curious 🤓
Numbers of the form \(2^{2^n}+1\) (for \(n \in \mathbb{Z}_{\ge 0}\)) are called Fermat numbers. There are many open problems around them:
  • Are there infinitely many Fermat primes?

  • Are there infinitely many composite Fermat numbers?

  • Does a Fermat number exist that is not square-free?

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