The problem
Fermat numbers are \(F_k\) = \(2^{2^k}\) + 1. \(F_0\) through \(F_4\) are prime (3, 5, 17, 257, 65537). Question: are there infinitely many Fermat primes? Or equivalently: does \(F_k\) produce a prime for any k > 4?
The Existence of Infinitely Many Fermat Primes
openHistory & significance
Fermat conjectured in 1640 that ALL Fermat numbers are prime. Euler demolished this in 1732 by finding 641 | \(F_5\). Since then, every \(F_5\) through \(F_32\) has been shown composite (some requiring massive distributed computation). Heuristics suggest only finitely many exist (expected total: ~5-6 based on probabilistic models), yet no proof rules out more. Each new candidate requires testing numbers with millions of digits.
Connected problems
More in number theory
- same area Dickson's Conjecture
- same area Schinzel's Hypothesis H
- same area The Fermat–Catalan Conjecture
References
- sourceWikipedia — Fermat number
- sourceWikipedia — Mersenne prime
Still open.
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