Fermat Primes (original) (raw)
Last Updated : 23 Jul, 2025
Fermat numbers are a special sequence of integers defined by the formula:
Fn = 2^{2^n} + 1
Where n is a non-negative integer. If any Fermat number is prime it is called Fermat Prime. Currently, there are **five known Fermat primes. These are:
- F0 = 2^{2^0} + 1 = 3
- F1 = 2^{2^1} + 1 = 5
- F2 = 2^{2^2} + 1 = 17
- F3 = 2^{2^3} + 1 = 257
- F4 = 2^{2^4} + 1 = 65537
Beyond **F 4 all known Fermat numbers Fn for n ≥5 [F5 = 4294967297], have been found to be composite numbers. The search for new Fermat Prime continues, but none has been discovered since **F 4.
**Note: all other Fermat numbers are divisible by some other numbers i.e.,
- F5 = 4294967297 (641 × 6700417)
- F6 = 18446744073709551617 (274177 × 67280421310721)
- F7 = 340282366920938463463374607431768211457 (59649589127497217 ×5704689200685129054721)
Facts about Fermat Primes
Some facts about Fermat number and primes are:
- All Fermat prime are **odd.
- There are only five known Fermat primes: 3, 5, 17, 257, and 65537.
- Fermat numbers are related to the construction of regular polygons. Specifically, a regular polygon with Fn sides can be constructed with a compass and straightedge if Fn is prime.
- Any two Fermat numbers Fm and Fn (where m ≠ n) are coprime, meaning gcd(Fm, Fn) = 1.
- 2^{(2^n)}+1 is a Fermat prime if and only if the period length of 1/(2^{(2^n)}+1) is equal to 2^{(2^n)}. In other words, Fermat primes are full reptend primes.
Conclusion
In conclusion, Fermat primes are a special type of prime number that follow a unique formula, F_n = 2^{2^n} + 1. Although Fermat once believed that all numbers in this form would be prime, we now know that only the first five are Fermat primes, and the rest are composite.
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