Fermat Numbers: Are All Numbers of the Form 2^(2ⁿ) + 1 Prime?

Fermat made a bold claim about a special family of numbers — numbers of the form 2^(2ⁿ) + 1, now known as Fermat numbers. He believed these numbers always turn out to be prime, no matter which non-negative integer n you choose. This conjecture captivated mathematicians for decades and sparked one of the most fascinating debates in number theory.

Fermat’s Conjecture About Fermat Numbers

Fermat stated:

Two raised to a power that is itself a power of two, when increased by one, always yields a prime number.

In other words, he claimed that every number of the form 2^(2ⁿ) + 1 is prime.

Fermat spent considerable time attempting to prove this in the general case. For the first several values of n, he provided explicit examples:

• F₁ = 2² + 1 = 5
• F₂ = 2⁴ + 1 = 17
• F₃ = 2⁸ + 1 = 257
• F₄ = 2¹⁶ + 1 = 65 537

Each of these is indeed prime.

Are All Fermat Numbers Really Prime?

So here is the question: do you think all numbers of this form are prime?