Fermat Primes

Fermat numbers are a special sequence of integers defined by the formula:

F_n =

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 =
• F₁ =
• F₂ =
• F₃ =
• F₄ =

Beyond F₄ all known Fermat numbers F_n 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 F_n ​sides can be constructed with a compass and straightedge if F_n​ is prime.
• Any two Fermat numbers F_m​ and F_n​ (where m ≠ n) are coprime, meaning gcd(F_m, F_n) = 1.
• is a Fermat prime if and only if the period length of is equal to . 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, . 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.

Read More,

• Prime Numbers
• Cullen Primes
• Repunit Primes