Proofs of Existence and Optimality · Difficult Problems in Number Theory (Optional)
Lesson 2
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 \(n\).
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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?
Number task, Advanced problem